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DOI: 10.4230/LIPIcs.WG.2026.2

Optimal b-Colourings and Fall Colourings in H-Free Graphs

Authors: Jungho Ahn, Tala Eagling-Vose, Felicia Lucke, David Manlove, Fabricio Mendoza Granada, and Daniël Paulusma

Published in: LIPIcs, Volume 376, 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)

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

DOI: 10.4230/LIPIcs.WG.2026.16

Colouring Graphs Without a Subdivided H-Graph: A Full Complexity Classification

Authors: Tala Eagling-Vose, Jorik Jooken, Felicia Lucke, Barnaby Martin, and Daniël Paulusma

Published in: LIPIcs, Volume 376, 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)

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

DOI: 10.4230/LIPIcs.ISAAC.2025.4

Finding d-Cuts in Claw-Free Graphs

Authors: Jungho Ahn, Tala Eagling-Vose, Felicia Lucke, Daniël Paulusma, and Siani Smith

Published in: LIPIcs, Volume 359, 36th International Symposium on Algorithms and Computation (ISAAC 2025)

Abstract

The Matching Cut problem is to decide if the vertex set of a connected graph can be partitioned into two non-empty sets B and R such that the edges between B and R form a matching, that is, every vertex in B has at most one neighbour in R, and vice versa. If for some integer d ≥ 1, we allow every vertex in B to have at most d neighbours in R, and vice versa, we obtain the more general problem d-Cut. It is known that d-Cut is NP-complete for every d ≥ 1. However, for claw-free graphs, it is only known that d-Cut is polynomial-time solvable for d = 1 and NP-complete for d ≥ 3. We resolve the missing case d = 2 by proving NP-completeness. This follows from our more general study, in which we also bound the maximum degree. That is, we prove that for every d ≥ 2, d-Cut, restricted to claw-free graphs of maximum degree p, is constant-time solvable if p ≤ 2d+1 and NP-complete if p ≥ 2d+3. Moreover, in the former case, we can find a d-cut in linear time. We also show how our positive results for claw-free graphs can be generalized to S_{1^t,𝓁}-free graphs where S_{1^t,𝓁} is the graph obtained from a star on t+2 vertices by subdividing one of its edges exactly 𝓁 times.

Cite as

Jungho Ahn, Tala Eagling-Vose, Felicia Lucke, Daniël Paulusma, and Siani Smith. Finding d-Cuts in Claw-Free Graphs. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 4:1-4:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{ahn_et_al:LIPIcs.ISAAC.2025.4,
  author =	{Ahn, Jungho and Eagling-Vose, Tala and Lucke, Felicia and Paulusma, Dani\"{e}l and Smith, Siani},
  title =	{{Finding d-Cuts in Claw-Free Graphs}},
  booktitle =	{36th International Symposium on Algorithms and Computation (ISAAC 2025)},
  pages =	{4:1--4:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-408-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{359},
  editor =	{Chen, Ho-Lin and Hon, Wing-Kai and Tsai, Meng-Tsung},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2025.4},
  URN =		{urn:nbn:de:0030-drops-249121},
  doi =		{10.4230/LIPIcs.ISAAC.2025.4},
  annote =	{Keywords: matching cut, d-cut, claw-free, maximum degree}
}

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Documents authored by Eagling-Vose, Tala

Optimal b-Colourings and Fall Colourings in H-Free Graphs

Abstract

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Colouring Graphs Without a Subdivided H-Graph: A Full Complexity Classification

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Finding d-Cuts in Claw-Free Graphs

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