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LIPIcs, Volume 376
52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)
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- published at: 2026-07-02
- Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- ISBN: 978-3-95977-430-7
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Complete Volume
LIPIcs, Volume 376, WG 2026, Complete Volume
Abstract
LIPIcs, Volume 376, WG 2026, Complete Volume
Cite as
52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 1-552, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@Proceedings{goedgebeur_et_al:LIPIcs.WG.2026,
title = {{LIPIcs, Volume 376, WG 2026, Complete Volume}},
booktitle = {52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)},
pages = {1--552},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-430-7},
ISSN = {1868-8969},
year = {2026},
volume = {376},
editor = {Goedgebeur, Jan and Rz\k{a}\.{z}ewski, Pawe{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WG.2026},
URN = {urn:nbn:de:0030-drops-269591},
doi = {10.4230/LIPIcs.WG.2026},
annote = {Keywords: LIPIcs, Volume 376, WG 2026, Complete Volume}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization
Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as
52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 0:i-0:xiv, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{goedgebeur_et_al:LIPIcs.WG.2026.0,
author = {Goedgebeur, Jan and Rz\k{a}\.{z}ewski, Pawe{\l}},
title = {{Front Matter, Table of Contents, Preface, Conference Organization}},
booktitle = {52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)},
pages = {0:i--0:xiv},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-430-7},
ISSN = {1868-8969},
year = {2026},
volume = {376},
editor = {Goedgebeur, Jan and Rz\k{a}\.{z}ewski, Pawe{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WG.2026.0},
URN = {urn:nbn:de:0030-drops-269587},
doi = {10.4230/LIPIcs.WG.2026.0},
annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
Revisiting Token Sliding on Chordal Graphs
Abstract
In this article, we revisit the complexity of the reconfiguration of independent sets under the token sliding rule on chordal graphs. In the Token Sliding Connectivity problem, the input is a graph G and an integer k, and the objective is to determine whether the reconfiguration graph TS_k(G) of G is connected. The vertices of TS_k(G) are k-independent sets of G, and two vertices are adjacent if and only if one can transform one of the two corresponding independent sets into the other by sliding a vertex (also called a token) along an edge. Bonamy and Bousquet [WG'17] proved that the Token Sliding Connectivity problem is polynomial-time solvable on interval graphs but NP-hard on split graphs. In light of these two results, the authors asked: can we decide the connectivity of TS_k(G) in polynomial time for chordal graphs with maximum clique-tree degree d? We answer this question in the negative and prove that the problem is NP-hard even when d = 4. We then study the parameterized complexity of the problem for a larger parameter called leafage and prove that the problem is co-W[1]-hard. We prove similar results for a closely related problem called Token Sliding Reachability. In this problem, the input is a graph G with two of its k-independent sets I and J, and the objective is to determine whether there is a sequence of valid token sliding moves that transform I into J.
Cite as
Rajat Adak, Saraswati Girish Nanoti, and Prafullkumar Tale. Revisiting Token Sliding on Chordal Graphs. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 1:1-1:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{adak_et_al:LIPIcs.WG.2026.1,
author = {Adak, Rajat and Nanoti, Saraswati Girish and Tale, Prafullkumar},
title = {{Revisiting Token Sliding on Chordal Graphs}},
booktitle = {52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)},
pages = {1:1--1:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-430-7},
ISSN = {1868-8969},
year = {2026},
volume = {376},
editor = {Goedgebeur, Jan and Rz\k{a}\.{z}ewski, Pawe{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WG.2026.1},
URN = {urn:nbn:de:0030-drops-261678},
doi = {10.4230/LIPIcs.WG.2026.1},
annote = {Keywords: Independent Set, Token Sliding, Chordal Graphs, Leafage, W\lbrack1\rbrack-hardness}
}
Document
Optimal b-Colourings and Fall Colourings in H-Free Graphs
Abstract
In a colouring of a graph, a vertex is b-chromatic if it is adjacent to a vertex of every other colour. We consider four well-studied colouring problems: b-Chromatic Number, Tight b-Chromatic Number, Fall Chromatic Number and Fall Achromatic Number, which fit into a framework based on whether every colour class has (i) at least one b-chromatic vertex, (ii) exactly one b-chromatic vertex, or (iii) all of its vertices being b-chromatic. By combining known and new results, we fully classify the computational complexity of b-Chromatic Number, Fall Chromatic Number and Fall Achromatic Number in H-free graphs. For Tight b-Chromatic Number in H-free graphs, we develop a general technique to determine new graphs H, for which the problem is polynomial-time solvable, and we also determine new graphs H, for which the problem is still NP-complete. We show, for the first time, the existence of a graph H such that in H-free graphs, b-Chromatic Number is NP-hard, while Tight b-Chromatic Number is polynomial-time solvable.
Cite as
Jungho Ahn, Tala Eagling-Vose, Felicia Lucke, David Manlove, Fabricio Mendoza Granada, and Daniël Paulusma. Optimal b-Colourings and Fall Colourings in H-Free Graphs. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 2:1-2:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{ahn_et_al:LIPIcs.WG.2026.2,
author = {Ahn, Jungho and Eagling-Vose, Tala and Lucke, Felicia and Manlove, David and Mendoza Granada, Fabricio and Paulusma, Dani\"{e}l},
title = {{Optimal b-Colourings and Fall Colourings in H-Free Graphs}},
booktitle = {52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)},
pages = {2:1--2:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-430-7},
ISSN = {1868-8969},
year = {2026},
volume = {376},
editor = {Goedgebeur, Jan and Rz\k{a}\.{z}ewski, Pawe{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WG.2026.2},
URN = {urn:nbn:de:0030-drops-261685},
doi = {10.4230/LIPIcs.WG.2026.2},
annote = {Keywords: b-chromatic number, tight graph, fall achromatic number, fall chromatic number, H-free graph}
}
Document
Cycles in Unions of Transitive Tournaments
Abstract
A tournament is an orientation of a complete graph. A bitournament is an orientation of a complete bipartite graph. Let G be an oriented graph which is the (not necessarily disjoint) union of a bounded number of transitive tournaments. When is it possible to remove the arc sets of a bounded number of transitive bitournaments from G in order to make G acyclic?
In this paper, we begin investigating this question and its ties to lettericity of graphs and geometric griddability of permutations, two independently developed notions that highlight very similar structural features in their respective objects. We explore the problem through the lens of "minimal obstructions", i.e. minimal classes of directed graphs for which this is not possible, and identify an infinite collection of such classes.
Cite as
Bogdan Alecu, Pedro Bureo Villafana, and Vadim Lozin. Cycles in Unions of Transitive Tournaments. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 3:1-3:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{alecu_et_al:LIPIcs.WG.2026.3,
author = {Alecu, Bogdan and Bureo Villafana, Pedro and Lozin, Vadim},
title = {{Cycles in Unions of Transitive Tournaments}},
booktitle = {52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)},
pages = {3:1--3:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-430-7},
ISSN = {1868-8969},
year = {2026},
volume = {376},
editor = {Goedgebeur, Jan and Rz\k{a}\.{z}ewski, Pawe{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WG.2026.3},
URN = {urn:nbn:de:0030-drops-261694},
doi = {10.4230/LIPIcs.WG.2026.3},
annote = {Keywords: Structural graph theory, directed graphs, permutation patterns, graph lettericity, minimum feedback arc set problem, well-quasi-orderability, transitive tournaments}
}
Document
Complexity of Firefighting on Graphs
Abstract
We consider a pursuit-evasion game that describes the process of extinguishing a fire burning on the nodes of an undirected graph. We denote the minimum number of firefighters required by ffn(G) and provide almost sharp bounds to this graph parameter for complete binary trees. We show that deciding whether ffn(G) ≤ m for given G and m is NP-hard. Furthermore, we show that shortest strategies can have superpolynomial length, leaving open whether the problem is in NP. We provide a construction that allows for transferring these results to a well-established Cops and Robbers variant called the "Hunter and Rabbit game".
Cite as
Julius Althoetmar, Jamico Schade, and Torben Schürenberg. Complexity of Firefighting on Graphs. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 4:1-4:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{althoetmar_et_al:LIPIcs.WG.2026.4,
author = {Althoetmar, Julius and Schade, Jamico and Sch\"{u}renberg, Torben},
title = {{Complexity of Firefighting on Graphs}},
booktitle = {52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)},
pages = {4:1--4:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-430-7},
ISSN = {1868-8969},
year = {2026},
volume = {376},
editor = {Goedgebeur, Jan and Rz\k{a}\.{z}ewski, Pawe{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WG.2026.4},
URN = {urn:nbn:de:0030-drops-261707},
doi = {10.4230/LIPIcs.WG.2026.4},
annote = {Keywords: Complexity, Cops and Robbers, Pursuit-Evasion}
}
Document
The S-Hamiltonian Cycle Problem
Abstract
Determining if an input undirected graph is Hamiltonian, i.e., if it has a cycle that visits every vertex exactly once, is one of the most famous NP-complete problems. We consider the following generalization of Hamiltonian cycles: for a fixed set S of natural numbers, we want to visit each vertex of a graph G exactly once and ensure that any two consecutive vertices can be joined in k hops for some choice of k ∈ S. Formally, an S-Hamiltonian cycle is a permutation (v₀,…,v_{n-1}) of the vertices of G such that, for 0 ≤ i ≤ n-1, there exists a walk between v_i and v_{i+1 mod n} whose length is in S. (We do not impose any constraints on how many times vertices can be visited as intermediate vertices of walks.) Of course Hamiltonian cycles in the standard sense correspond to S = {1}. We study the S-Hamiltonian cycle problem of deciding whether an input graph G has an S-Hamiltonian cycle. Our goal is to determine the complexity of this problem depending on the fixed set S. It is already known that the problem remains NP-complete for S = {1,2}, whereas it is trivial for S = {1,2,3} because any connected graph contains a {1,2,3}-Hamiltonian cycle.
Our work classifies the complexity of this problem for most kinds of sets S, with the key new results being the following: we have NP-completeness for S = {2} and for S = {2, 4}, but tractability for S = {1, 2, 4}, for S = {2, 4, 6}, for any superset of these two tractable cases, and for S the infinite set of all odd integers. The remaining open cases are the non-singleton finite sets of odd integers, in particular S = {1, 3}. Beyond cycles, we also discuss the complexity of finding S-Hamiltonian paths, and show that our problems are all tractable on graphs of bounded cliquewidth.
Cite as
Antoine Amarilli, Arthur Lombardo, and Mikaël Monet. The S-Hamiltonian Cycle Problem. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 5:1-5:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{amarilli_et_al:LIPIcs.WG.2026.5,
author = {Amarilli, Antoine and Lombardo, Arthur and Monet, Mika\"{e}l},
title = {{The S-Hamiltonian Cycle Problem}},
booktitle = {52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)},
pages = {5:1--5:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-430-7},
ISSN = {1868-8969},
year = {2026},
volume = {376},
editor = {Goedgebeur, Jan and Rz\k{a}\.{z}ewski, Pawe{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WG.2026.5},
URN = {urn:nbn:de:0030-drops-261711},
doi = {10.4230/LIPIcs.WG.2026.5},
annote = {Keywords: Graph, Cycle, Hamiltonian}
}
Document
Upward-Planar Drawings with Bounded Span
Abstract
We consider upward-planar layered drawings of directed graphs, i.e., crossing-free drawings in which each edge is drawn as a y-monotone curve going upward from its tail to its head, and the y-coordinates of the vertices are integers. The span of an edge in such a drawing is the absolute difference between the y-coordinates of its endpoints, and the span of the drawing is the maximum span of any edge. The span of an upward-planar graph is the minimum span over all its upward-planar drawings.
We study the problem of determining the span of upward-planar graphs and provide both combinatorial and algorithmic results. On the combinatorial side, we present upper and lower bounds for the span of directed trees. On the algorithmic side, we show that the problem of determining the span of an upward-planar graph is NP-complete already for directed trees and for biconnected single-source graphs. Moreover, we give efficient algorithms for several graph families with a bounded number of sources, including st-planar graphs and graphs where the planar or upward-planar embedding is prescribed. Furthermore, we show that the problem is fixed-parameter tractable with respect to the vertex cover number and the treedepth plus the span.
Cite as
Patrizio Angelini, Sabine Cornelsen, Giordano Da Lozzo, Fabrizio Frati, Philipp Kindermann, Ignaz Rutter, and Johannes Zink. Upward-Planar Drawings with Bounded Span. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 6:1-6:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{angelini_et_al:LIPIcs.WG.2026.6,
author = {Angelini, Patrizio and Cornelsen, Sabine and Da Lozzo, Giordano and Frati, Fabrizio and Kindermann, Philipp and Rutter, Ignaz and Zink, Johannes},
title = {{Upward-Planar Drawings with Bounded Span}},
booktitle = {52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)},
pages = {6:1--6:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-430-7},
ISSN = {1868-8969},
year = {2026},
volume = {376},
editor = {Goedgebeur, Jan and Rz\k{a}\.{z}ewski, Pawe{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WG.2026.6},
URN = {urn:nbn:de:0030-drops-261722},
doi = {10.4230/LIPIcs.WG.2026.6},
annote = {Keywords: graph drawing, directed graphs, upward planarity, span, level drawings}
}
Document
Making an Oriented Graph Acyclic Using Inversions of Bounded or Prescribed Size
Abstract
Given an oriented graph D, the inversion of a subset X of vertices consists in reversing the orientation of all arcs with both endpoints in X. When the subset X is of size p (resp. at most p), this operation is called an (= p)-inversion (resp. (⩽ p)-inversion). Then, an oriented graph is (= p)-invertible if it can be made acyclic by a sequence of p-inversions. We observe that, for n = |V(D)|, deciding whether D is (= n-1)-invertible is equivalent to deciding whether D is acyclically pushable, and thus NP-complete. In all other cases, whenever p ≠ n-1, we construct a polynomial-time algorithm deciding (= p)-invertibility.
We then consider the (= p)-inversion number, inv^{= p}(D) (resp. (⩽ p)-inversion number, inv^{⩽ p}(D)), defined as the minimum number of (= p)-inversions (resp. (⩽ p)-inversions) rendering D acyclic. We show that every (= p)-invertible digraph D satisfies inv^{= p}(D) ⩽ |A(D)| for every integer p ⩾ 2. When p is even, we moreover bound inv^{= p} by a (linear) function of the feedback arc set number, and rule out the existence of any bounding function for odd p.
Finally, we study the complexity of deciding whether the (= p)-inversion number, or the (⩽ p)-inversion number, of a given oriented graph is at most a given integer k. For any fixed positive integer p ⩾ 2, when k is part of the input, we show that both problems are NP-hard even in tournaments. In general oriented graphs, we prove W[1]-hardness for both problems when parameterized by p, even for k = 1. In contrast, we exhibit polynomial kernels in p + k for both problems in tournaments.
Cite as
Jørgen Bang-Jensen, Frédéric Havet, Florian Hörsch, Clément Rambaud, Amadeus Reinald, and Caroline Silva. Making an Oriented Graph Acyclic Using Inversions of Bounded or Prescribed Size. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 7:1-7:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bangjensen_et_al:LIPIcs.WG.2026.7,
author = {Bang-Jensen, J{\o}rgen and Havet, Fr\'{e}d\'{e}ric and H\"{o}rsch, Florian and Rambaud, Cl\'{e}ment and Reinald, Amadeus and Silva, Caroline},
title = {{Making an Oriented Graph Acyclic Using Inversions of Bounded or Prescribed Size}},
booktitle = {52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)},
pages = {7:1--7:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-430-7},
ISSN = {1868-8969},
year = {2026},
volume = {376},
editor = {Goedgebeur, Jan and Rz\k{a}\.{z}ewski, Pawe{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WG.2026.7},
URN = {urn:nbn:de:0030-drops-261733},
doi = {10.4230/LIPIcs.WG.2026.7},
annote = {Keywords: digraph, inversion, orientation, NP-hardness, acyclic, reconfiguration}
}
Document
Tight Bounds for Some W[1]-Hard Problems Parameterized by Multi-Clique-Width
Abstract
In this work we contribute to the study of the fine-grained complexity of problems parameterized by multi-clique-width, which was initiated by Fürer [ITCS 2017] and pursued further by Chekan and Kratsch [MFCS 2023]. Multi-clique-width is a parameter defined analogously to clique-width but every vertex is allowed to hold multiple labels simultaneously. This parameter is upper-bounded by both clique-width and treewidth (plus a constant), hence it generalizes both of them without an exponential blow-up. Conversely, graphs of multi-clique-width k have clique-width at most 2^k, and there exist graphs with clique-width at least 2^{Ω(k)}. Thus, while the two parameters are functionally equivalent, the fine-grained complexity of problems may differ relative to them.
As our first and main result we show that under ETH the Max Cut problem cannot be solved in time n^{2^{o(k)}} ⋅ f(k) on graphs of multi-clique-width k for any computable function f. For clique-width k an n^{𝒪(k)} algorithm by Fomin et al. [SIAM J. Comput. 2014] is tight under ETH. This makes Max Cut the first known problem for which the tight running times differ for parameterization by clique-width and multi-clique-width and it contributes to the short list of known lower bounds of form n^{2^{o(k)}} ⋅ f(k). As our second contribution we show that Hamiltonian Cycle and Edge Dominating Set can be solved in time n^{𝒪(k)} on graphs of multi-clique-width k matching the tight running time for clique-width. These results answer three questions left open by Chekan and Kratsch [MFCS 2023].
Cite as
Benjamin Bergougnoux, Vera Chekan, and Stefan Kratsch. Tight Bounds for Some W[1]-Hard Problems Parameterized by Multi-Clique-Width. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 8:1-8:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bergougnoux_et_al:LIPIcs.WG.2026.8,
author = {Bergougnoux, Benjamin and Chekan, Vera and Kratsch, Stefan},
title = {{Tight Bounds for Some W\lbrack1\rbrack-Hard Problems Parameterized by Multi-Clique-Width}},
booktitle = {52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)},
pages = {8:1--8:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-430-7},
ISSN = {1868-8969},
year = {2026},
volume = {376},
editor = {Goedgebeur, Jan and Rz\k{a}\.{z}ewski, Pawe{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WG.2026.8},
URN = {urn:nbn:de:0030-drops-261741},
doi = {10.4230/LIPIcs.WG.2026.8},
annote = {Keywords: Parameterized complexity, multi-clique-width, tight bounds, ETH}
}
Document
Trade-Off Between Spread and Width for Tree Decompositions
Abstract
We study the trade-off between (average) spread and width in tree decompositions, answering several questions from Wood [arXiv:2509.01140]. The spread of a vertex v in a tree decomposition is the number of bags that contain v. Wood asked for which c > 0, there exists c' such that each graph G has a tree decomposition of width ctw(G) in which each vertex v has spread at most c'(d(v)+1). We show that c ≥ 2 is necessary and that c > 3 is sufficient. Moreover, we answer a second question fully by showing that near-optimal average spread can be achieved simultaneously with width O(tw(G)).
Cite as
Hans L. Bodlaender and Carla Groenland. Trade-Off Between Spread and Width for Tree Decompositions. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 9:1-9:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bodlaender_et_al:LIPIcs.WG.2026.9,
author = {Bodlaender, Hans L. and Groenland, Carla},
title = {{Trade-Off Between Spread and Width for Tree Decompositions}},
booktitle = {52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)},
pages = {9:1--9:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-430-7},
ISSN = {1868-8969},
year = {2026},
volume = {376},
editor = {Goedgebeur, Jan and Rz\k{a}\.{z}ewski, Pawe{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WG.2026.9},
URN = {urn:nbn:de:0030-drops-261753},
doi = {10.4230/LIPIcs.WG.2026.9},
annote = {Keywords: Tree decomposition, spread, domino treewidth}
}
Document
Implicit Representations via the Polynomial Method
Abstract
Semialgebraic graphs are graphs whose vertices are points in {ℝ}^d, and adjacency between two vertices is determined by the truth value of a semialgebraic predicate of constant complexity. We show how to harness polynomial partitioning methods to construct compact adjacency labeling schemes for families of semialgebraic graphs.
That is, we show that for any family of semialgebraic graphs, given a graph on n vertices in this family, we can assign a label consisting of O(n^{1-2/(d+1) + {ε}}) bits to each vertex (where {ε} > 0 can be made arbitrarily small and the constant of proportionality depends on {ε} and on the complexity of the adjacency-defining predicate), such that adjacency between two vertices can be determined solely from their two labels, without any additional information. We obtain for instance that unit disk graphs and segment intersection graphs have such labelings with labels of O(n^{1/3 + {ε}}) bits. This is in contrast to their natural implicit representation consisting of the coordinates of the disk centers or segment endpoints, which sometimes require exponentially many bits. It also improves on the best known bound of O(n^{1-1/d}log n) for d-dimensional semialgebraic families due to Alon (Discrete Comput. Geom., 2024), a bound that holds more generally for graphs with shattering functions bounded by a degree-d polynomial. Our labeling scheme is efficient in the sense that not only adjacency between two vertices can be decided in time linear in the size of their labels, but the labels can be computed in subquadratic time on a real RAM from the input points and the semialgebraic adjacency predicate, using recent polynomial partitioning algorithms.
We also give new bounds on the size of adjacency labels for other families of graphs. In particular, we consider semilinear graphs, which are semialgebraic graphs in which the predicate only involves linear polynomials. We show that semilinear graphs have adjacency labels of size O(log n). We also prove that polygon visibility graphs, which are not semialgebraic in the above sense, have adjacency labels of size O(log³ n).
Cite as
Jean Cardinal and Micha Sharir. Implicit Representations via the Polynomial Method. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 10:1-10:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{cardinal_et_al:LIPIcs.WG.2026.10,
author = {Cardinal, Jean and Sharir, Micha},
title = {{Implicit Representations via the Polynomial Method}},
booktitle = {52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)},
pages = {10:1--10:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-430-7},
ISSN = {1868-8969},
year = {2026},
volume = {376},
editor = {Goedgebeur, Jan and Rz\k{a}\.{z}ewski, Pawe{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WG.2026.10},
URN = {urn:nbn:de:0030-drops-261762},
doi = {10.4230/LIPIcs.WG.2026.10},
annote = {Keywords: Semialgebraic graphs, Geometric intersection graphs, Visibility graphs, Adjacency labelings, Polynomial partitioning}
}
Document
Parameterized Complexity of Isometric Path Partition: Treewidth and Diameter
Abstract
In the Isometric Path Partition problem, the input is a graph G with n vertices and an integer k, and the objective is to determine whether the vertices of G can be partitioned into k vertex-disjoint shortest paths. We investigate the parameterized complexity of the problem when parameterized by the treewidth (tw) of the input graph, arguably one of the most widely studied parameters. Courcelle’s theorem [Information & Computation, 1990] shows that graph problems that are expressible as MSO formulas of constant size admit FPT algorithms parameterized by the treewidth of the input graph. This encompasses many natural graph problems. However, many metric-based graph problems, where the solution is defined using some metric-based property of the graph (often the distance) are not expressible as MSO formulas of constant size. These types of problems, Isometric Path Partition being one of them, require individual attention and often draw the boundary for the success story of parameterization by treewidth.
We show that Isometric Path Partition is W[1]-hard when parameterized by treewidth (in fact, even pathwidth (pw)), answering the question by Dumas et al. [SIDMA, 2024], Fernau et al. [TCS, 2025], and confirming the aforementioned tendency. We complement this hardness result by designing a tailored dynamic programming algorithm running in n^{O(tw)} time. This dynamic programming approach also results in an algorithm running in time diam^{O(tw²)} ⋅ n^{O(1)}, where diam is the diameter of the graph. It is known that Isometric Path Partition remains NP-hard on graphs of diameter 2; hence, the combination of both parameters is necessary to obtain a tractable algorithm. Note that the dependency on treewidth is unusually high, as most problems that are FPT for treewidth admit algorithms running in time 2^{O(tw)}⋅ n^{O(1)} or 2^{O(tw log (tw))}⋅ n^{O(1)}. However, we rule out the possibility of a significantly faster algorithm, showing that Isometric Path Partition does not admit an algorithm running in time diam^{o(pw²/(log³(pw)))} ⋅ n^{O(1)}, assuming the Randomized-ETH.
Cite as
Dibyayan Chakraborty, Oscar Defrain, Florent Foucaud, Mathieu Mari, and Prafullkumar Tale. Parameterized Complexity of Isometric Path Partition: Treewidth and Diameter. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 11:1-11:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{chakraborty_et_al:LIPIcs.WG.2026.11,
author = {Chakraborty, Dibyayan and Defrain, Oscar and Foucaud, Florent and Mari, Mathieu and Tale, Prafullkumar},
title = {{Parameterized Complexity of Isometric Path Partition: Treewidth and Diameter}},
booktitle = {52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)},
pages = {11:1--11:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-430-7},
ISSN = {1868-8969},
year = {2026},
volume = {376},
editor = {Goedgebeur, Jan and Rz\k{a}\.{z}ewski, Pawe{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WG.2026.11},
URN = {urn:nbn:de:0030-drops-261774},
doi = {10.4230/LIPIcs.WG.2026.11},
annote = {Keywords: Isometric path partition, parameterized complexity, treewidth, diameter, Randomized ETH}
}
Document
Geometric Routing in Geometric Inhomogeneous Random Graphs
Abstract
We present the first rigorous analysis of decentralized geometric routing in Geometric Inhomogeneous Random Graphs (GIRGs), a weight-agnostic variant of the greedy routing protocol. While greedy routing in GIRGs is known to explain the algorithmic small-world phenomenon by finding ultra-short paths of length Θ(log log n), it assumes additional knowledge of vertex weights beyond geometry, an assumption that is often restrictive or unavailable. We investigate whether the underlying geometry alone is sufficient for efficient navigation. We prove that for power-law weight exponent τ ∈ (2,3) and geometric decay parameter α > τ-1, geometric routing succeeds with constant probability and finds ultra-short paths of length Θ(log log n), matching the optimal asymptotic guarantees for greedy routing. Our analysis further reveals that, upon success, both protocols follow a similar two-phase trajectory, consisting of a rapid ascent to the heavy vertices, followed by efficient navigation to the target. These results demonstrate that, in the appropriate regime, the network’s geometry alone implicitly guides the path to the target through its high-weight core.
Cite as
Yu-Cheng Chiu, Marc Kaufmann, Kostas Lakis, and Ulysse Schaller. Geometric Routing in Geometric Inhomogeneous Random Graphs. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 12:1-12:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{chiu_et_al:LIPIcs.WG.2026.12,
author = {Chiu, Yu-Cheng and Kaufmann, Marc and Lakis, Kostas and Schaller, Ulysse},
title = {{Geometric Routing in Geometric Inhomogeneous Random Graphs}},
booktitle = {52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)},
pages = {12:1--12:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-430-7},
ISSN = {1868-8969},
year = {2026},
volume = {376},
editor = {Goedgebeur, Jan and Rz\k{a}\.{z}ewski, Pawe{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WG.2026.12},
URN = {urn:nbn:de:0030-drops-261780},
doi = {10.4230/LIPIcs.WG.2026.12},
annote = {Keywords: geometric inhomogeneous random graphs (GIRGs), hyperbolic random graphs (HRGs), greedy routing, geometric routing, navigability, small-world phenomenon, decentralized algorithms}
}
Document
Is Graph Local Complementation Inherently Sequential?
Abstract
Local complementation of a graph G on vertex v is an operation that results in a new graph G*v, where the neighborhood of v is complemented. Two graph are locally equivalent if one can be reached from the other one through local complementation.
It was previously established that recognizing locally equivalent graphs can be done in 𝒪(n⁴) time. We sharpen this result by proving it can be decided in 𝒪(log²(n)) parallel time with n^{𝒪(1)} processors.
As a second contribution, we introduce the Local Complementation Problem, a decision problem that captures the complexity of applying a sequence of local complementations. Given a graph G, a sequence of vertices s, and a pair of vertices u,v, the problem asks whether the edge (u,v) is present in the graph obtained after applying local complementations according to s. Despite its simplicity, it is proven to be {𝐏}-complete, therefore it is unlikely to be efficiently parallelizable.
Finally, it is conjectured that Local Complementation Problem remains {𝐏}-complete when restricted to circle graphs.
Cite as
Pablo Concha-Vega. Is Graph Local Complementation Inherently Sequential?. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 13:1-13:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{conchavega:LIPIcs.WG.2026.13,
author = {Concha-Vega, Pablo},
title = {{Is Graph Local Complementation Inherently Sequential?}},
booktitle = {52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)},
pages = {13:1--13:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-430-7},
ISSN = {1868-8969},
year = {2026},
volume = {376},
editor = {Goedgebeur, Jan and Rz\k{a}\.{z}ewski, Pawe{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WG.2026.13},
URN = {urn:nbn:de:0030-drops-261798},
doi = {10.4230/LIPIcs.WG.2026.13},
annote = {Keywords: Local complementation, P-completeness, vertex-minors, graph transformations}
}
Document
Graph Classes Closed Under Self-Intersection
Abstract
A graph class is monotone if it is closed under taking subgraphs. A monotone class defined by finitely many obstructions has bounded treewidth if and only if one of the obstructions is a tripod, i.e. a disjoint union of subdivided claws and paths. This dichotomy also characterizes exactly those monotone graph classes for which many NP-hard graph problems admit polynomial-time algorithms. These dichotomies do not extend to the universe of all hereditary classes. This leads to the question of whether we can extend known dichotomies for monotone classes to larger families of hereditary classes. We answer this question affirmatively by considering the family of hereditary graph classes closed under self-intersection. This family is known to be located strictly between the monotone and hereditary classes. We prove a new structural characterization of graphs in self-intersection-closed classes excluding a tripod. In contrast to monotone classes excluding a tripod, these classes do not necessarily have bounded treewidth; in fact, they do not even need to be sparse. We use our characterization to give a complete dichotomy for Maximum Independent Set, and its weighted variant, on self-intersection-closed classes defined by finitely many obstructions: these problems are in P if the class excludes a tripod and NP-hard otherwise. Our dichotomy generalizes several known results on Maximum Independent Set in the literature. We also apply our characterization to obtain a dichotomy for Maximum Induced Matching on self-intersection-closed classes of bipartite graphs defined by finitely many obstructions, and for Satisfiability and Counting Satisfiability on self-intersection-closed classes of (bipartite) incidence graphs defined by finitely many obstructions. Finally, we use our characterization to obtain a dichotomy for boundedness of clique-width for self-intersection-closed classes of bipartite graphs defined by finitely many obstructions.
Cite as
Konrad K. Dabrowski, Vadim Lozin, Martin Milanič, Andrea Munaro, Daniël Paulusma, and Viktor Zamaraev. Graph Classes Closed Under Self-Intersection. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 14:1-14:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{dabrowski_et_al:LIPIcs.WG.2026.14,
author = {Dabrowski, Konrad K. and Lozin, Vadim and Milani\v{c}, Martin and Munaro, Andrea and Paulusma, Dani\"{e}l and Zamaraev, Viktor},
title = {{Graph Classes Closed Under Self-Intersection}},
booktitle = {52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)},
pages = {14:1--14:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-430-7},
ISSN = {1868-8969},
year = {2026},
volume = {376},
editor = {Goedgebeur, Jan and Rz\k{a}\.{z}ewski, Pawe{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WG.2026.14},
URN = {urn:nbn:de:0030-drops-261801},
doi = {10.4230/LIPIcs.WG.2026.14},
annote = {Keywords: graph classes, self-intersection closed, dichotomy, independent set, clique-width, treewidth}
}
Document
High Beer Index Implies Big Hollow Triangles
Abstract
The visibility graph of a set S ⊆ ℝ² is the graph whose vertices are the points of S, with two points x,y connected by an edge if and only if they see each other in S, that is, if the segment xy is contained in S. The edge density of this graph is known as the Beer index of S.
Previously, it has been shown that a simply connected set S ⊆ ℝ² of unit Lebesgue measure with Beer index β > 0 contains a convex subset of measure Ω(β); in particular, for visibility graphs of simply connected sets, a positive edge density β > 0 implies the existence of a clique containing an Ω(β)-fraction of all vertices. The simple-connectivity assumption cannot be omitted, as there are non-simply-connected sets with Beer index 1 and no convex subset of positive measure.
Nevertheless, in this paper, we extend the above result to non-simply-connected sets, by showing that a visibility graph with large edge density contains a triangle with large convex hull. More precisely, we show that a set S ⊆ ℝ² of unit Lebesgue measure with Beer index β > 0 contains three pairwise visible points whose convex hull has measure Ω(β⁹). If in addition S is an open domain with K holes, then S contains three pairwise visible points with convex hull of measure Ω(β/K) as well as a convex subset of measure Ω(β/K²).
Cite as
Arun Kumar Das, Vít Jelínek, Jan Kynčl, Martin Pergel, Felix Schröder, Peter Stumpf, and Pavel Valtr. High Beer Index Implies Big Hollow Triangles. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 15:1-15:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{das_et_al:LIPIcs.WG.2026.15,
author = {Das, Arun Kumar and Jel{\'\i}nek, V{\'\i}t and Kyn\v{c}l, Jan and Pergel, Martin and Schr\"{o}der, Felix and Stumpf, Peter and Valtr, Pavel},
title = {{High Beer Index Implies Big Hollow Triangles}},
booktitle = {52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)},
pages = {15:1--15:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-430-7},
ISSN = {1868-8969},
year = {2026},
volume = {376},
editor = {Goedgebeur, Jan and Rz\k{a}\.{z}ewski, Pawe{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WG.2026.15},
URN = {urn:nbn:de:0030-drops-261813},
doi = {10.4230/LIPIcs.WG.2026.15},
annote = {Keywords: convexity, Beer index, visibility graph}
}
Document
Colouring Graphs Without a Subdivided H-Graph: A Full Complexity Classification
Abstract
We consider Colouring on graphs that are H-subgraph-free for some fixed graph H, which are graphs that do not contain H as a subgraph. To classify the complexity of Colouring on H-subgraph-free graphs for connected H, it remains to consider when H is a tree of maximum degree 4 with exactly one vertex of degree 4, or a tree of maximum degree 3 with at least two vertices of degree 3. We let H be a so-called subdivided "H"-graph, which is either a subdivided ℍ₀: a tree of maximum degree 4 that is a star, or a subdivided ℍ₁: a tree of maximum degree 3 with exactly two vertices of degree 3. We develop new decomposition theorems resulting in polynomial-time algorithms, and in combination with known results, fully classify all cases ℍ₀ and ℍ₁. To illustrate the wider applicability of our techniques, we also employ them to obtain similar new polynomial-time results for two other classic graph problems: Stable Cut and, in part, Feedback Vertex Set.
Cite as
Tala Eagling-Vose, Jorik Jooken, Felicia Lucke, Barnaby Martin, and Daniël Paulusma. Colouring Graphs Without a Subdivided H-Graph: A Full Complexity Classification. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 16:1-16:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{eaglingvose_et_al:LIPIcs.WG.2026.16,
author = {Eagling-Vose, Tala and Jooken, Jorik and Lucke, Felicia and Martin, Barnaby and Paulusma, Dani\"{e}l},
title = {{Colouring Graphs Without a Subdivided H-Graph: A Full Complexity Classification}},
booktitle = {52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)},
pages = {16:1--16:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-430-7},
ISSN = {1868-8969},
year = {2026},
volume = {376},
editor = {Goedgebeur, Jan and Rz\k{a}\.{z}ewski, Pawe{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WG.2026.16},
URN = {urn:nbn:de:0030-drops-261827},
doi = {10.4230/LIPIcs.WG.2026.16},
annote = {Keywords: colouring, forbidden subgraph, complexity dichotomy}
}
Document
Clustering with Locally Bounded Ignorance
Abstract
In Correlation Clustering, the input is a graph G = (V,E) with weight function ω: {V choose 2} → ℤ_{≥ 0} and the task is to partition the vertex set into clusters such that the total weight of edges between clusters and missing edges inside clusters is minimized. Due to close connections between Correlation Clustering and Edge Multicut, deciding whether there is a partition with total cost at most k is FPT with respect to k but a polynomial kernel is presumably impossible. We study the influence of the structure of the fuzzy edge graph, that is, the graph induced by the weight-0 edges, on the problem complexity. We show in particular that Correlation Clustering admits a polynomial problem kernel when parameterized by k+d, where d is the degeneracy of the fuzzy edge graph, and when parameterized by k+c, where c is the closure of the fuzzy edge graph. We complement these positive results by showing hardness for several settings where the graph induced by the edges and nonedges has very restricted structure.
Cite as
Jaroslav Garvardt and Christian Komusiewicz. Clustering with Locally Bounded Ignorance. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 17:1-17:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{garvardt_et_al:LIPIcs.WG.2026.17,
author = {Garvardt, Jaroslav and Komusiewicz, Christian},
title = {{Clustering with Locally Bounded Ignorance}},
booktitle = {52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)},
pages = {17:1--17:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-430-7},
ISSN = {1868-8969},
year = {2026},
volume = {376},
editor = {Goedgebeur, Jan and Rz\k{a}\.{z}ewski, Pawe{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WG.2026.17},
URN = {urn:nbn:de:0030-drops-261834},
doi = {10.4230/LIPIcs.WG.2026.17},
annote = {Keywords: Graph-based Data Clustering, Cluster Editing, Kernelization}
}
Document
Polynomial Kernels for Spanning Tree with Diversity Requirements
Abstract
Given a connected undirected graph G, a spanning tree is a subgraph T of G such that V(T) = V(G) and T is a tree. A collection of 𝓁 spanning trees T₁,…,T_{𝓁} is {{pairwise k-diverse}} if for every i ≠ j, |E(T_i) △ E(T_j)| ≥ k. Given a connected undirected graph G and integers p, q, k, 𝓁, {Leaf&Internal-Constrained Diverse Spanning Trees} asks whether there are 𝓁 distinct spanning trees T₁,…,T_{𝓁} of G that are {{pairwise k-diverse}} such that each tree has at least p leaves and at least q internal vertices. Similarly, {Leaf&Non-terminal-Constrained Diverse Spanning Trees} takes a connected undirected graph G, V_NT ⊆ V(G), and three integers p, k, 𝓁, and asks if G has 𝓁 spanning trees that are {{pairwise k-diverse}}, and each has at least p leaves and contains the vertices of V_NT as internal. We consider these two problems from the kernelization perspective and provide polynomial kernels for {Leaf&Internal-Constrained Diverse Spanning Trees} and {Leaf&Non-terminal-Constrained Diverse Spanning Trees}, when parameterized by p + q + k + 𝓁 and p + |V_NT| + k + 𝓁, respectively.
Cite as
Petr A. Golovach, Diptapriyo Majumdar, and Saket Saurabh. Polynomial Kernels for Spanning Tree with Diversity Requirements. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{golovach_et_al:LIPIcs.WG.2026.18,
author = {Golovach, Petr A. and Majumdar, Diptapriyo and Saurabh, Saket},
title = {{Polynomial Kernels for Spanning Tree with Diversity Requirements}},
booktitle = {52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)},
pages = {18:1--18:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-430-7},
ISSN = {1868-8969},
year = {2026},
volume = {376},
editor = {Goedgebeur, Jan and Rz\k{a}\.{z}ewski, Pawe{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WG.2026.18},
URN = {urn:nbn:de:0030-drops-261840},
doi = {10.4230/LIPIcs.WG.2026.18},
annote = {Keywords: Parameterized Complexity, Kernelization, Diverse Solutions, Diverse Spanning Trees}
}
Document
Faster 3-Colouring Algorithm for Graphs of Diameter 3
Abstract
We show that given an n-vertex graph G of diameter 3 we can decide if G is 3-colourable in time 2^{O(n^{2/3-ε})} for any ε < 1/33. This improves on the previous best algorithm of 2^{O((nlog n)^{2/3})} from Dębski, Piecyk and Rzążewski [Faster 3-coloring of small-diameter graphs, ESA 2021].
Cite as
Carla Groenland, Hidde Koerts, and Sophie Spirkl. Faster 3-Colouring Algorithm for Graphs of Diameter 3. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 19:1-19:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{groenland_et_al:LIPIcs.WG.2026.19,
author = {Groenland, Carla and Koerts, Hidde and Spirkl, Sophie},
title = {{Faster 3-Colouring Algorithm for Graphs of Diameter 3}},
booktitle = {52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)},
pages = {19:1--19:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-430-7},
ISSN = {1868-8969},
year = {2026},
volume = {376},
editor = {Goedgebeur, Jan and Rz\k{a}\.{z}ewski, Pawe{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WG.2026.19},
URN = {urn:nbn:de:0030-drops-261851},
doi = {10.4230/LIPIcs.WG.2026.19},
annote = {Keywords: 3-colouring, subexponential-time algorithm, diameter-3 graphs}
}
Document
Computational and Combinatorial Results on Conflict-Free Choosability
Abstract
The conflict-free closed neighborhood (CFCN^*) chromatic number of a graph G = (V,E) is the smallest positive integer k for which there exists a coloring of a subset of vertices using k colors such that, for every vertex in V, there exists a color that appears exactly once in its closed neighborhood. The conflict-free open neighborhood (CFON^*) chromatic number is defined analogously. In this paper, we study "list variants" of the above-mentioned coloring parameters. The conflict-free closed neighborhood (CFCN^*) choice number of a graph G = (V,E) is the smallest positive integer k such that for every assignment of lists of size k to its vertices, there exists a coloring of a subset of vertices, say V', in which (i) every vertex in V' receives a color from its list, and (ii) for every vertex in V there exists some color that appears exactly once in its closed neighborhood. The conflict-free open neighborhood (CFON^*) choice number is defined analogously.
Dębski and Przybyło [Journal of Graph Theory, 2022] showed that for any graph G with maximum degree Δ, the CFCN^* chromatic number of its line graph is O(ln Δ). This result was later extended to claw-free graphs by Bhyravarapu et al. [Journal of Graph Theory, 2023], who proved that every K_{1,k}-free graph G admits a CFCN^* coloring using O(kln Δ) colors. In this paper, we generalize this result to the list setting and show that every K_{1,k}-free graph G has a CFCN^* choice number of O(kln Δ). Further, we answer some questions concerning the hardness of computing CFCN^*/CFON^* choice numbers posed by Gupta and Mathew [SOFSEM, 2026]; in particular, we show that it is NP-hard to determine whether the CFCN^*/CFON^* choice number a graph is equal to k, for k = 1,2.
Cite as
Shiwali Gupta and Rogers Mathew. Computational and Combinatorial Results on Conflict-Free Choosability. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 20:1-20:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{gupta_et_al:LIPIcs.WG.2026.20,
author = {Gupta, Shiwali and Mathew, Rogers},
title = {{Computational and Combinatorial Results on Conflict-Free Choosability}},
booktitle = {52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)},
pages = {20:1--20:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-430-7},
ISSN = {1868-8969},
year = {2026},
volume = {376},
editor = {Goedgebeur, Jan and Rz\k{a}\.{z}ewski, Pawe{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WG.2026.20},
URN = {urn:nbn:de:0030-drops-261868},
doi = {10.4230/LIPIcs.WG.2026.20},
annote = {Keywords: conflict-free coloring, list conflict-free coloring, choice number, claw number, computational complexity, hardness results}
}
Document
Hunting for Directed 2-Spiders
Abstract
Hons, Klimošová, Mikšaník, Tkadlec, Tyomkyn and the second author proved that, for every integer 𝓁 ≥ 1, every directed graph with minimum out-degree at least 3.23 ⋅ 𝓁 contains a (2,𝓁)-spider (a 1-subdivision of the in-star with 𝓁 leaves) as a subgraph. Hons et al. also conjectured that the bound on the minimum out-degree can be further improved to 2 𝓁. In this note, we confirm this conjecture by showing that every directed graph with minimum out-degree at least 2𝓁 contains a (2, 𝓁)-spider as a subgraph. This result is best possible, as the complete directed graph with 2𝓁 vertices does not contain a (2,𝓁)-spider.
Cite as
Grzegorz Gutowski and Gaurav Kucheriya. Hunting for Directed 2-Spiders. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 21:1-21:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{gutowski_et_al:LIPIcs.WG.2026.21,
author = {Gutowski, Grzegorz and Kucheriya, Gaurav},
title = {{Hunting for Directed 2-Spiders}},
booktitle = {52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)},
pages = {21:1--21:6},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-430-7},
ISSN = {1868-8969},
year = {2026},
volume = {376},
editor = {Goedgebeur, Jan and Rz\k{a}\.{z}ewski, Pawe{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WG.2026.21},
URN = {urn:nbn:de:0030-drops-261879},
doi = {10.4230/LIPIcs.WG.2026.21},
annote = {Keywords: Oriented and Directed Graphs, Extremal Graph Theory, Mathematics of Computing, Unavoidable Subgraphs}
}
Document
A Note on the Complexity of Directed Clique
Abstract
For a directed graph G, and a linear order ≪ on the vertices of G, we define the backedge graph G^≪ to be the undirected graph on the same vertex set with edge {u,w} in G^≪ if and only if (u,w) is an arc in G and w ≪ u. The directed clique number of a directed graph G is defined as the minimum size of the maximum clique in the backedge graph G^≪ taken over all linear orders ≪ on the vertices of G. A natural computational problem is to decide for a given directed graph G and a positive integer t, if the directed clique number of G is at most t. This problem has polynomial algorithm for t = 1 and is known to be NP-complete for every fixed t ≥ 3, even for tournaments. In this note we prove that this problem is Σ^𝖯₂-complete when t is given on the input.
Cite as
Grzegorz Gutowski and Mikołaj Rams. A Note on the Complexity of Directed Clique. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 22:1-22:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{gutowski_et_al:LIPIcs.WG.2026.22,
author = {Gutowski, Grzegorz and Rams, Miko{\l}aj},
title = {{A Note on the Complexity of Directed Clique}},
booktitle = {52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)},
pages = {22:1--22:10},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-430-7},
ISSN = {1868-8969},
year = {2026},
volume = {376},
editor = {Goedgebeur, Jan and Rz\k{a}\.{z}ewski, Pawe{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WG.2026.22},
URN = {urn:nbn:de:0030-drops-261885},
doi = {10.4230/LIPIcs.WG.2026.22},
annote = {Keywords: Directed Clique, Computational Complexity, Polynomial Hierarchy}
}
Document
Parameterized Complexity of Power Network Design: Coordinating Cable Placement Is Hard
Abstract
We study several generalizations of the Steiner Tree problem that are motivated by the design of power networks. While Steiner Tree asks for a single minimum-cost tree that connects a given set of terminal vertices, a power network typically consists of multiple trees. Each tree connects to a subset of the terminals, to avoid electrical overloads. The cost of installing a power network is therefore determined by two factors: the total length of the cables in the network and the cost of digging underground trenches into which the cables are placed. Since the digging costs can be substantial, to minimize the total cost of the network it might be necessary to place multiple cables into the same trench. These characteristics lead to variations of Steiner Tree in which the goal is to compute a minimum-cost set of Steiner trees, all with a common root, that together connect a given terminal set while balancing the power demand of the terminals in each tree. Two important variations arise depending on whether the network is intended for low-voltage or high-voltage power. In the low-voltage setting, there is substantial power loss across the cables which effectively means that the maximum depth of any tree in the solution has to be bounded. No such depth bound applies to the high-voltage setting.
We investigate the parameterized complexity of several power network design problems, using the number of terminals as the parameter. While this parameterization of the standard Steiner Tree problem is fixed-parameter tractable, many of our variants are W[1]-hard. For low-voltage networks (bounded-depth trees), we present an XP-algorithm for planar inputs, which exploits a nontrivial bound on the treewidth of solution subgraphs. We provide an intricate reduction from Grid Tiling to establish that the resulting algorithm is tight under the Exponential Time Hypothesis. The XP-algorithm extends to the high-voltage setting and to general graphs, albeit at a cost in the running time. For high-voltage networks, we prove that the problem remains W[1]-hard on planar graphs. Finally, we explore a variation of the cost model for sharing digging costs in which both problems become fixed-parameter tractable.
Cite as
Thekla Hamm, Bart M. P. Jansen, and Faezeh Motiei. Parameterized Complexity of Power Network Design: Coordinating Cable Placement Is Hard. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 23:1-23:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{hamm_et_al:LIPIcs.WG.2026.23,
author = {Hamm, Thekla and Jansen, Bart M. P. and Motiei, Faezeh},
title = {{Parameterized Complexity of Power Network Design: Coordinating Cable Placement Is Hard}},
booktitle = {52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)},
pages = {23:1--23:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-430-7},
ISSN = {1868-8969},
year = {2026},
volume = {376},
editor = {Goedgebeur, Jan and Rz\k{a}\.{z}ewski, Pawe{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WG.2026.23},
URN = {urn:nbn:de:0030-drops-261892},
doi = {10.4230/LIPIcs.WG.2026.23},
annote = {Keywords: Steiner Tree, Network Design, Parameterized Complexity, ETH-tight Algorithm}
}
Document
Graph Reconstruction with a Connected Components Oracle
Abstract
In the Graph Reconstruction (GR) problem, the goal is to recover a hidden graph by utilizing some oracle that provides limited access to the structure of the graph. The interest is in characterizing how strong different oracles are when the complexity of an algorithm is measured in the number of performed queries. We study a novel oracle that returns the set of connected components (CC) on the subgraph induced by the queried subset of vertices. Our main contributions are as follows:
1) For a hidden graph with n vertices, m edges, maximum degree Δ, and treewidth k, GR can be solved in 𝒪(min{m/log m, Δ², k²} ⋅ log n) CC queries by an adaptive randomized algorithm.
2) For a hidden graph with n vertices and degeneracy d, GR can be solved in 𝒪(d² log² n) CC queries by an adaptive randomized algorithm.
3) There are hidden graphs with n vertices, m edges, maximum degree Δ, treewidth k, and degeneracy d such that Ω(m), Ω(Δ²), Ω(k²), and Ω(d²) CC queries are required for solving GR.
Cite as
Juha Harviainen and Pekka Parviainen. Graph Reconstruction with a Connected Components Oracle. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 24:1-24:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{harviainen_et_al:LIPIcs.WG.2026.24,
author = {Harviainen, Juha and Parviainen, Pekka},
title = {{Graph Reconstruction with a Connected Components Oracle}},
booktitle = {52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)},
pages = {24:1--24:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-430-7},
ISSN = {1868-8969},
year = {2026},
volume = {376},
editor = {Goedgebeur, Jan and Rz\k{a}\.{z}ewski, Pawe{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WG.2026.24},
URN = {urn:nbn:de:0030-drops-261907},
doi = {10.4230/LIPIcs.WG.2026.24},
annote = {Keywords: graph reconstruction, parameterized complexity, query complexity}
}
Document
The Complexity of Ramsey Arrowing: A Computational Approach for Hardness Proofs
Abstract
In graph Ramsey theory, the arrowing operator is used to describe the appearance of unavoidable substructures within colored graphs; for graphs G, F, and H, we say G → (F,H) (read, G arrows F, H) if every red/blue coloring of G’s edges contains a red F or a blue H. For fixed F and H, the (F,H)-Arrowing problem asks whether G → (F,H) for some given graph G. (F,H)-Arrowing has been shown to be in P or coNP-complete for different pairs (F,H). However, categorizing the complexity for all pairs still remains wide open.
In general, categorizing the complexity of problems whose nature depends on some underlying graph - or, in our case, pair of graphs - is a daunting task, and (F,H)-Arrowing is no exception. Hardness proofs typically rely on ad-hoc, laborious constructions of special graphs known as "gadgets." In this work, we present a simple, computational approach to find these gadgets for small (F,H)-Arrowing problems and show how these can be extended to other (F,H)-Arrowing problems.
Our main focus is on the simplest case for which the complexity remains uncategorized: F = P₃. We showcase the efficacy of our computational approach by presenting hardness proofs for (P₃, H)-Arrowing problems previously not known to be coNP-hard. Moreover, we show how to generalize hardness to other (P₃,H)-Arrowing problems by either: (1) carefully inspecting and modifying our found gadgets, or (2) coming up with intuitive constructions to reduce (P₃, H')-Arrowing to (P₃, H)-Arrowing, where H' is a subgraph of H. We also discuss how our methodology can be extended to work for other (F,H)-Arrowing problems by showing new results for F = P₄ and K_{1,3}.
We see our work as an important step towards categorizing the complexity of (F,H)-Arrowing for all pairs (F,H). (F,H)-Arrowing is thought to be hard when (F',H')-Arrowing is hard where F' and H' are subgraphs of F and H, respectively. Under this assumption, our results narrow down the only uncategorized family of problems for which the problem may lie in P. Beyond the complexity of (F,H)-Arrowing, we believe that the broader impact of our work is showing how computational methodologies can be adopted for proving hardness, and we hope our approach will be adopted to find gadgets for other open graph problems as well.
Cite as
Zohair Raza Hassan. The Complexity of Ramsey Arrowing: A Computational Approach for Hardness Proofs. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 25:1-25:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{hassan:LIPIcs.WG.2026.25,
author = {Hassan, Zohair Raza},
title = {{The Complexity of Ramsey Arrowing: A Computational Approach for Hardness Proofs}},
booktitle = {52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)},
pages = {25:1--25:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-430-7},
ISSN = {1868-8969},
year = {2026},
volume = {376},
editor = {Goedgebeur, Jan and Rz\k{a}\.{z}ewski, Pawe{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WG.2026.25},
URN = {urn:nbn:de:0030-drops-261916},
doi = {10.4230/LIPIcs.WG.2026.25},
annote = {Keywords: Graph Arrowing, Ramsey theory, Hardness reductions, Computational proofs}
}
Document
Weisfeiler-Leman on Graphs of Small Twin-Width
Abstract
Twin-width is a graph parameter introduced in the context of first-order model checking, and has since become a central parameter in algorithmic graph theory. While many algorithmic problems become easier on arbitrary classes of bounded twin-width, graph isomorphism on graphs of twin-width 4 and above is as hard as the general isomorphism problem. For each positive integer k, the k-dimensional Weisfeiler-Leman algorithm is an iterative color refinement algorithm that encodes structural similarities and serves as a fundamental tool for distinguishing non-isomorphic graphs. We show that the graph isomorphism problem for graphs of twin-width 1 can be solved by the 3-dimensional Weisfeiler-Leman algorithm, while there is no fixed k such that the k-dimensional Weisfeiler-Leman algorithm solves the graph isomorphism problem for graphs of twin-width 4. Moreover, we prove the conjecture of Bergougnoux, Gajarský, Guspiel, Hlinený, Pokrývka, and Sokolowski (ISAAC 2023) that stable graphs of twin-width 2 have bounded rank-width. This implies that isomorphism of these graphs is solved by a fixed dimension of the Weisfeiler-Leman algorithm.
Cite as
Irene Heinrich, Moritz Lichter, Klara Pakhomenko, and Simon Raßmann. Weisfeiler-Leman on Graphs of Small Twin-Width. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 26:1-26:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{heinrich_et_al:LIPIcs.WG.2026.26,
author = {Heinrich, Irene and Lichter, Moritz and Pakhomenko, Klara and Ra{\ss}mann, Simon},
title = {{Weisfeiler-Leman on Graphs of Small Twin-Width}},
booktitle = {52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)},
pages = {26:1--26:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-430-7},
ISSN = {1868-8969},
year = {2026},
volume = {376},
editor = {Goedgebeur, Jan and Rz\k{a}\.{z}ewski, Pawe{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WG.2026.26},
URN = {urn:nbn:de:0030-drops-261929},
doi = {10.4230/LIPIcs.WG.2026.26},
annote = {Keywords: twin-width, Weisfeiler-Leman algorithm, canonization, half-graph, rank-width}
}
Document
The Planar Edge-Coloring Theorem of Vizing in O(nlog n) Time
Abstract
In 1965, Vizing [Vadim G. Vizing, 1965] showed that every planar graph of maximum degree Δ ≥ 8 can be edge-colored using Δ colors. The direct implementation of Vizing’s proof gives an algorithm that finds the coloring in O(n²) time for an n-vertex input graph. Chrobak and Nishizeki [Marek Chrobak and Takao Nishizeki, 1990] have shown a more careful algorithm, which improves the time to O(nlog n), though only for Δ ≥ 9. In this paper, we extend their ideas to get an algorithm also for the missing case Δ = 8. To this end, we modify the original recoloring procedure of Vizing. This generalizes to bounded genus graphs of maximum degree 8 in the sense that in time O(nlog n) the algorithm colors the graph using the optimal number of colors which may be 9 for relatively small graphs.
Cite as
Patryk Jędrzejczak and Łukasz Kowalik. The Planar Edge-Coloring Theorem of Vizing in O(nlog n) Time. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 27:1-27:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{jedrzejczak_et_al:LIPIcs.WG.2026.27,
author = {J\k{e}drzejczak, Patryk and Kowalik, {\L}ukasz},
title = {{The Planar Edge-Coloring Theorem of Vizing in O(nlog n) Time}},
booktitle = {52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)},
pages = {27:1--27:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-430-7},
ISSN = {1868-8969},
year = {2026},
volume = {376},
editor = {Goedgebeur, Jan and Rz\k{a}\.{z}ewski, Pawe{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WG.2026.27},
URN = {urn:nbn:de:0030-drops-261931},
doi = {10.4230/LIPIcs.WG.2026.27},
annote = {Keywords: edge coloring, algorithm, planar, graph, Vizing, quasilinear}
}
Document
Lower Bounds for the Pfaffian Number of Graphs
Abstract
The number of perfect matchings of a k-pfaffian graph can be counted by computing a linear combination of the pfaffians of k matrices. The pfaffian number of a graph G is the smallest integer k such that G is k-pfaffian. We present the first known lower bounds for the pfaffian number of graphs. As an intermediate step, we prove an upper bound for the rank of two matrices related to their Khatri-Rao product. One of the consequences of the found lower bounds is the existence of graphs whose pfaffian numbers are arbitrarily large.
Cite as
Enrique Junchaya, Alberto Alexandre Assis Miranda, and Cláudio Leonardo Lucchesi. Lower Bounds for the Pfaffian Number of Graphs. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 28:1-28:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{junchaya_et_al:LIPIcs.WG.2026.28,
author = {Junchaya, Enrique and Assis Miranda, Alberto Alexandre and Lucchesi, Cl\'{a}udio Leonardo},
title = {{Lower Bounds for the Pfaffian Number of Graphs}},
booktitle = {52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)},
pages = {28:1--28:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-430-7},
ISSN = {1868-8969},
year = {2026},
volume = {376},
editor = {Goedgebeur, Jan and Rz\k{a}\.{z}ewski, Pawe{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WG.2026.28},
URN = {urn:nbn:de:0030-drops-261946},
doi = {10.4230/LIPIcs.WG.2026.28},
annote = {Keywords: Perfect matchings, pfaffian graphs, k-pfaffian graphs, pfaffian number of graphs}
}
Document
Obstructions for Minor-Closed Classes of Limiting Densities Below 3/2
Abstract
Given a graph class 𝒢, the limiting density of 𝒢 is defined as δ(𝒢) = lim_{n → ∞} ex(𝒢,n)/n where ex(𝒢,n) is the maximum number of edges of a graph in 𝒢 on n vertices. The limiting density δ(𝒢) is known to be a rational number when 𝒢 is a minor-closed graph class. For every δ ∈ [0,3/2), we prove that the set of ⊆-minimal minor-closed graph classes with densities > δ is finite and we identify it completely. A consequence of our results is an algorithm that, given a finite set of graphs 𝒵, of total size n, either outputs the value of δ(excl(𝒵)) or reports that δ(excl(𝒵)) ≥ 3/2, where excl(𝒵) is the class of graphs excluding the graphs in 𝒵 as minors. The algorithm runs in 2^{poly(n)} time.
Cite as
Antonios Kominatos, Reem Mahmoud, and Dimitrios M. Thilikos. Obstructions for Minor-Closed Classes of Limiting Densities Below 3/2. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 29:1-29:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kominatos_et_al:LIPIcs.WG.2026.29,
author = {Kominatos, Antonios and Mahmoud, Reem and Thilikos, Dimitrios M.},
title = {{Obstructions for Minor-Closed Classes of Limiting Densities Below 3/2}},
booktitle = {52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)},
pages = {29:1--29:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-430-7},
ISSN = {1868-8969},
year = {2026},
volume = {376},
editor = {Goedgebeur, Jan and Rz\k{a}\.{z}ewski, Pawe{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WG.2026.29},
URN = {urn:nbn:de:0030-drops-261952},
doi = {10.4230/LIPIcs.WG.2026.29},
annote = {Keywords: Graph Minors, Limiting density, Obstruction set, Class property, Parametric graph}
}
Document
Preventing Small Global Cuts by Protecting Edges
Abstract
The minimum cut problem is one of the oldest and most fundamental optimization problems in operations research. In this problem, we are given a connected edge-weighted graph (G,ω) and have to find an edge set A (called edge-cut) of smallest total weight such that the removal of the edges of A disconnects G. The problem thus takes the view of an attacker that wants to destroy the global connectivity of the network. Bienstock and Diaz [SICOMP '93] introduced Global Cut Prevention, a two-player version of the minimum cut problem where a defender aims to protect edges to increase the weight of the minimum cut of the resulting graph. More precisely, the input contains an additional edge cost function c that is independent of the attacker weight ω and the defender aims to protect an edge set of total cost at most d such that every edge-cut consisting of unprotected edges has weight at least a+1. We initiate the study of the parameterized complexity of Global Cut Prevention. Here, we consider the most natural parameters such as the budgets d and a of the players, the vertex cover number and treewidth of the input graph, and combinations of these parameters. We show, for example, that the encoding of the costs and weights of the edges has a considerable influence on the problem complexity: If each edge has unit defender cost and unit attacker weight, then Global Cut Prevention is FPT for the vertex cover number. If the attacker weights are arbitrary and encoded in unary, then the problem is W[1]-hard for the vertex cover number but still admits an XP-algorithm. Finally, if the defender cost and the attacker weight are encoded in binary, then the problem becomes NP-hard even on graphs with a vertex cover of size 2.
Cite as
Christian Komusiewicz, Zhenwei Liu, Nils Morawietz, and Frank Sommer. Preventing Small Global Cuts by Protecting Edges. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 30:1-30:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{komusiewicz_et_al:LIPIcs.WG.2026.30,
author = {Komusiewicz, Christian and Liu, Zhenwei and Morawietz, Nils and Sommer, Frank},
title = {{Preventing Small Global Cuts by Protecting Edges}},
booktitle = {52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)},
pages = {30:1--30:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-430-7},
ISSN = {1868-8969},
year = {2026},
volume = {376},
editor = {Goedgebeur, Jan and Rz\k{a}\.{z}ewski, Pawe{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WG.2026.30},
URN = {urn:nbn:de:0030-drops-261964},
doi = {10.4230/LIPIcs.WG.2026.30},
annote = {Keywords: Network interdiction, NP-hard problem, parameterized complexity, structural parameterization, edge-weighted graphs}
}
Document
On the Relation Between Treewidth, Tree-Independence Number, and Tree-Chromatic Number of Graphs
Abstract
We investigate the relationship between graph parameters, which measure the complexity of the tree decompositions of a given graph. The treewidth tw(G) of a graph G measures the largest number of vertices required in a bag of every tree decomposition of G. Similarly, the tree-independence number tree-α(G) and the tree-chromatic number tree-χ(G) measure the largest independence number, respectively the largest chromatic number, required in a bag of every tree decomposition of G. Recently, Dallard, Milanič, and Štorgel asked (JCTB, 2024) whether for all graphs G it holds that tw(G)+1 ≤ tree-α(G) ⋅ tree-χ(G). We provide a negative answer for this question in a strong form: for every function f: {ℕ} → {ℕ}, there exists a graph G such that tw(G) > tree-α(G) ⋅ f(tree-χ(G)). On the other hand, we complement this result with an upper bound, by showing that tw(G)+1 ≤ tree-α(G)² ⋅ tree-χ(G) for every graph G.
Cite as
Alex Koutsoutis, Kilian Krause, Chun-Hung Liu, Mirza Redzic, and Torsten Ueckerdt. On the Relation Between Treewidth, Tree-Independence Number, and Tree-Chromatic Number of Graphs. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 31:1-31:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{koutsoutis_et_al:LIPIcs.WG.2026.31,
author = {Koutsoutis, Alex and Krause, Kilian and Liu, Chun-Hung and Redzic, Mirza and Ueckerdt, Torsten},
title = {{On the Relation Between Treewidth, Tree-Independence Number, and Tree-Chromatic Number of Graphs}},
booktitle = {52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)},
pages = {31:1--31:9},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-430-7},
ISSN = {1868-8969},
year = {2026},
volume = {376},
editor = {Goedgebeur, Jan and Rz\k{a}\.{z}ewski, Pawe{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WG.2026.31},
URN = {urn:nbn:de:0030-drops-261978},
doi = {10.4230/LIPIcs.WG.2026.31},
annote = {Keywords: Tree-independence number, Tree-chromatic number, Treewidth}
}
Document
Optimal Path Partitions in Subcubic and Almost-Subcubic Graphs
Abstract
We consider the problem of partitioning the edges of a graph into as few paths as possible. This is a subject of the classic conjecture of Gallai and a recurring topic in combinatorics. Regarding the complexity of partitioning a graph optimally, Peroché [Discrete Appl. Math. 1984] proved that it is NP-hard already on graphs of maximum degree four, even when we only ask if two paths suffice.
We show that the problem is solvable in polynomial time on subcubic graphs and then we present an efficient algorithm for "almost-subcubic" graphs. Precisely, we prove that the problem is fixed-parameter tractable when parameterized by the edge-deletion distance to a subcubic graph. To this end, we reduce the task to model checking in first-order logic extended by disjoint-paths predicates (FO+DP) and then we employ the recent tractability result by Schirrmacher, Siebertz, Stamoulis, Thilikos, and Vigny [LICS 2024].
Cite as
Tomáš Masařík, Michał Włodarczyk, and Mehmet Akif Yıldız. Optimal Path Partitions in Subcubic and Almost-Subcubic Graphs. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 32:1-32:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{masarik_et_al:LIPIcs.WG.2026.32,
author = {Masa\v{r}{\'\i}k, Tom\'{a}\v{s} and W{\l}odarczyk, Micha{\l} and Y{\i}ld{\i}z, Mehmet Akif},
title = {{Optimal Path Partitions in Subcubic and Almost-Subcubic Graphs}},
booktitle = {52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)},
pages = {32:1--32:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-430-7},
ISSN = {1868-8969},
year = {2026},
volume = {376},
editor = {Goedgebeur, Jan and Rz\k{a}\.{z}ewski, Pawe{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WG.2026.32},
URN = {urn:nbn:de:0030-drops-261987},
doi = {10.4230/LIPIcs.WG.2026.32},
annote = {Keywords: path partitions, parameterized algorithms, subcubic graphs, model checking, disjoint paths}
}
Document
Totally Δ-Modular Tree Decompositions of Graphic Matrices for Integer Programming
Abstract
We introduce the tree-decomposition-based parameter totally Δ-modular treewidth (TDM-treewidth) for matrices with two nonzero entries per row. We show how to solve integer programs whose matrices have bounded TDM-treewidth in polynomial time when variables have bounded domain. This extends previous graph-based decomposition parameters for matrices with at most two nonzero entries per row to include matrices with entries outside of {-1,0,1}. We also give an analogue of the Grid Theorem of Robertson and Seymour for matrices of bounded TDM-treewidth in the language of rooted signed graphs.
Cite as
Caleb McFarland. Totally Δ-Modular Tree Decompositions of Graphic Matrices for Integer Programming. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 33:1-33:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{mcfarland:LIPIcs.WG.2026.33,
author = {McFarland, Caleb},
title = {{Totally \Delta-Modular Tree Decompositions of Graphic Matrices for Integer Programming}},
booktitle = {52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)},
pages = {33:1--33:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-430-7},
ISSN = {1868-8969},
year = {2026},
volume = {376},
editor = {Goedgebeur, Jan and Rz\k{a}\.{z}ewski, Pawe{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WG.2026.33},
URN = {urn:nbn:de:0030-drops-261992},
doi = {10.4230/LIPIcs.WG.2026.33},
annote = {Keywords: Integer programming, subdeterminants, independent set, rooted graphs, signed graphs, odd cycle packing number}
}