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**Published in:** LIPIcs, Volume 292, 3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024)

In this paper we initiate the study of the temporal graph realization problem with respect to the fastest path durations among its vertices, while we focus on periodic temporal graphs. Given an n × n matrix D and a Δ ∈ ℕ, the goal is to construct a Δ-periodic temporal graph with n vertices such that the duration of a fastest path from v_i to v_j is equal to D_{i,j}, or to decide that such a temporal graph does not exist. The variations of the problem on static graphs has been well studied and understood since the 1960’s (e.g. [Erdős and Gallai, 1960], [Hakimi and Yau, 1965]).
As it turns out, the periodic temporal graph realization problem has a very different computational complexity behavior than its static (i. e., non-temporal) counterpart. First we show that the problem is NP-hard in general, but polynomial-time solvable if the so-called underlying graph is a tree. Building upon those results, we investigate its parameterized computational complexity with respect to structural parameters of the underlying static graph which measure the "tree-likeness". We prove a tight classification between such parameters that allow fixed-parameter tractability (FPT) and those which imply W[1]-hardness. We show that our problem is W[1]-hard when parameterized by the feedback vertex number (and therefore also any smaller parameter such as treewidth, degeneracy, and cliquewidth) of the underlying graph, while we show that it is in FPT when parameterized by the feedback edge number (and therefore also any larger parameter such as maximum leaf number) of the underlying graph.

Nina Klobas, George B. Mertzios, Hendrik Molter, and Paul G. Spirakis. Temporal Graph Realization from Fastest Paths. In 3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 292, pp. 16:1-16:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{klobas_et_al:LIPIcs.SAND.2024.16, author = {Klobas, Nina and Mertzios, George B. and Molter, Hendrik and Spirakis, Paul G.}, title = {{Temporal Graph Realization from Fastest Paths}}, booktitle = {3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024)}, pages = {16:1--16:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-315-7}, ISSN = {1868-8969}, year = {2024}, volume = {292}, editor = {Casteigts, Arnaud and Kuhn, Fabian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2024.16}, URN = {urn:nbn:de:0030-drops-198945}, doi = {10.4230/LIPIcs.SAND.2024.16}, annote = {Keywords: Temporal graph, periodic temporal labeling, fastest temporal path, graph realization, temporal connectivity, parameterized complexity} }

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Brief Announcement

**Published in:** LIPIcs, Volume 292, 3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024)

We consider random simple temporal graphs in which every edge of the complete graph K_n appears once within the time interval [0,1] independently and uniformly at random. Our main result is a sharp threshold on the size of any maximum δ-clique (namely a clique with edges appearing at most δ apart within [0,1]) in random instances of this model, for any constant δ. In particular, using the probabilistic method, we prove that the size of a maximum δ-clique is approximately (2 log n)/(log 1/δ) with high probability (whp). What seems surprising is that, even though the random simple temporal graph contains Θ(n²) overlapping δ-windows, which (when viewed separately) correspond to different random instances of the Erdős-Rényi random graphs model, the size of the maximum δ-clique in the former model and the maximum clique size of the latter are approximately the same. Furthermore, we show that the minimum interval containing a δ-clique is δ-o(δ) whp. We use this result to show that any polynomial time algorithm for δ-Temporal Clique is unlikely to have very large probability of success.

George B. Mertzios, Sotiris Nikoletseas, Christoforos Raptopoulos, and Paul G. Spirakis. Brief Announcement: On the Existence of δ-Temporal Cliques in Random Simple Temporal Graphs. In 3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 292, pp. 27:1-27:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{mertzios_et_al:LIPIcs.SAND.2024.27, author = {Mertzios, George B. and Nikoletseas, Sotiris and Raptopoulos, Christoforos and Spirakis, Paul G.}, title = {{Brief Announcement: On the Existence of \delta-Temporal Cliques in Random Simple Temporal Graphs}}, booktitle = {3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024)}, pages = {27:1--27:5}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-315-7}, ISSN = {1868-8969}, year = {2024}, volume = {292}, editor = {Casteigts, Arnaud and Kuhn, Fabian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2024.27}, URN = {urn:nbn:de:0030-drops-199056}, doi = {10.4230/LIPIcs.SAND.2024.27}, annote = {Keywords: Simple random temporal graph, \delta-temporal clique, probabilistic method} }

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Invited Talk

**Published in:** LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

Graphs are fundamental tools for modelling relations among objects in various scientific fields. However, traditional static graphs have limitations when it comes to capturing the dynamic nature of real-world systems. To overcome this limitation, temporal graphs have been introduced as a framework to model graphs that change over time. In temporal graphs the edges among vertices appear and disappear at specific time steps, reflecting the temporal dynamics of the observed system, which allows us to analyse time dependent patterns and processes. In this paper we focus on the research related to sliding time windows in temporal graphs. Sliding time windows offer a way to analyse specific time intervals within the lifespan of a temporal graph. By sliding the window along the timeline, we can examine the graph’s characteristics and properties within different time periods.
This paper provides an overview of the research on sliding time windows in temporal graphs. Although progress has been made in this field, there are still many interesting questions and challenges to be explored. We discuss some of the open problems and highlight their potential for future research.

Nina Klobas, George B. Mertzios, and Paul G. Spirakis. Sliding into the Future: Investigating Sliding Windows in Temporal Graphs (Invited Talk). In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 5:1-5:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{klobas_et_al:LIPIcs.MFCS.2023.5, author = {Klobas, Nina and Mertzios, George B. and Spirakis, Paul G.}, title = {{Sliding into the Future: Investigating Sliding Windows in Temporal Graphs}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {5:1--5:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-292-1}, ISSN = {1868-8969}, year = {2023}, volume = {272}, editor = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.5}, URN = {urn:nbn:de:0030-drops-185397}, doi = {10.4230/LIPIcs.MFCS.2023.5}, annote = {Keywords: Temporal Graphs, Sliding Time Windows} }

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**Published in:** LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

A graph is temporally connected if there exists a strict temporal path, i.e., a path whose edges have strictly increasing labels, from every vertex u to every other vertex v. In this paper we study temporal design problems for undirected temporally connected graphs. The basic setting of these optimization problems is as follows: given a connected undirected graph G, what is the smallest number |λ| of time-labels that we need to add to the edges of G such that the resulting temporal graph (G,λ) is temporally connected? As it turns out, this basic problem, called Minimum Labeling (ML), can be optimally solved in polynomial time. However, exploiting the temporal dimension, the problem becomes more interesting and meaningful in its following variations, which we investigate in this paper. First we consider the problem Min. Aged Labeling (MAL) of temporally connecting the graph when we are given an upper-bound on the allowed age (i.e., maximum label) of the obtained temporal graph (G,λ). Second we consider the problem Min. Steiner Labeling (MSL), where the aim is now to have a temporal path between any pair of "important" vertices which lie in a subset R ⊆ V, which we call the terminals. This relaxed problem resembles the problem Steiner Tree in static (i.e., non-temporal) graphs. However, due to the requirement of strictly increasing labels in a temporal path, Steiner Tree is not a special case of MSL. Finally we consider the age-restricted version of MSL, namely Min. Aged Steiner Labeling (MASL). Our main results are threefold: we prove that (i) MAL becomes NP-complete on undirected graphs, while (ii) MASL becomes W[1]-hard with respect to the number |R| of terminals. On the other hand we prove that (iii) although the age-unrestricted problem MSL remains NP-hard, it is in FPT with respect to the number |R| of terminals. That is, adding the age restriction, makes the above problems strictly harder (unless P=NP or W[1]=FPT).

Nina Klobas, George B. Mertzios, Hendrik Molter, and Paul G. Spirakis. The Complexity of Computing Optimum Labelings for Temporal Connectivity. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 62:1-62:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{klobas_et_al:LIPIcs.MFCS.2022.62, author = {Klobas, Nina and Mertzios, George B. and Molter, Hendrik and Spirakis, Paul G.}, title = {{The Complexity of Computing Optimum Labelings for Temporal Connectivity}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {62:1--62:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-256-3}, ISSN = {1868-8969}, year = {2022}, volume = {241}, editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.62}, URN = {urn:nbn:de:0030-drops-168603}, doi = {10.4230/LIPIcs.MFCS.2022.62}, annote = {Keywords: Temporal graph, graph labeling, foremost temporal path, temporal connectivity, Steiner Tree} }

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**Published in:** Dagstuhl Reports, Volume 11, Issue 3 (2021)

This report documents the program and the outcomes of Dagstuhl Seminar 121171 "Temporal Graphs: Structure, Algorithms, Applications". The seminar was organized around four core areas: models, concepts, classes; concrete algorithmic problems; distributed aspects; applications. Because of the ongoing pandemic crisis, the seminar had to be held fully online, with talk and open problems sessions focussing on afternoons. Besides 19 contributed talks and small-group discussions, there were lively open-problem sessions, and some of the problems and research directions proposed there are part of this document. Despite strongly missing the usual Dagstuhl atmosphere and personal interaction possibilities, the seminar helped to establish new contacts and to identify new research directions in a thriving research area between (algorithmic) graph theory and network science.

Arnaud Casteigts, Kitty Meeks, George B. Mertzios, and Rolf Niedermeier. Temporal Graphs: Structure, Algorithms, Applications (Dagstuhl Seminar 21171). In Dagstuhl Reports, Volume 11, Issue 3, pp. 16-46, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@Article{casteigts_et_al:DagRep.11.3.16, author = {Casteigts, Arnaud and Meeks, Kitty and Mertzios, George B. and Niedermeier, Rolf}, title = {{Temporal Graphs: Structure, Algorithms, Applications (Dagstuhl Seminar 21171)}}, pages = {16--46}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2021}, volume = {11}, number = {3}, editor = {Casteigts, Arnaud and Meeks, Kitty and Mertzios, George B. and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.11.3.16}, URN = {urn:nbn:de:0030-drops-146892}, doi = {10.4230/DagRep.11.3.16}, annote = {Keywords: algorithm engineering, complex network analysis, distributed computing, models and classes, parameterized complexity analysis} }

Document

**Published in:** LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

In a temporal network with discrete time-labels on its edges, entities and information can only "flow" along sequences of edges whose time-labels are non-decreasing (resp. increasing), i.e. along temporal (resp. strict temporal) paths. Nevertheless, in the model for temporal networks of [Kempe, Kleinberg, Kumar, JCSS, 2002], the individual time-labeled edges remain undirected: an edge e = {u,v} with time-label t specifies that "u communicates with v at time t". This is a symmetric relation between u and v, and it can be interpreted that the information can flow in either direction. In this paper we make a first attempt to understand how the direction of information flow on one edge can impact the direction of information flow on other edges. More specifically, naturally extending the classical notion of a transitive orientation in static graphs, we introduce the fundamental notion of a temporal transitive orientation and we systematically investigate its algorithmic behavior in various situations. An orientation of a temporal graph is called temporally transitive if, whenever u has a directed edge towards v with time-label t₁ and v has a directed edge towards w with time-label t₂ ≥ t₁, then u also has a directed edge towards w with some time-label t₃ ≥ t₂. If we just demand that this implication holds whenever t₂ > t₁, the orientation is called strictly temporally transitive, as it is based on the fact that there is a strict directed temporal path from u to w. Our main result is a conceptually simple, yet technically quite involved, polynomial-time algorithm for recognizing whether a given temporal graph 𝒢 is transitively orientable. In wide contrast we prove that, surprisingly, it is NP-hard to recognize whether 𝒢 is strictly transitively orientable. Additionally we introduce and investigate further related problems to temporal transitivity, notably among them the temporal transitive completion problem, for which we prove both algorithmic and hardness results.

George B. Mertzios, Hendrik Molter, Malte Renken, Paul G. Spirakis, and Philipp Zschoche. The Complexity of Transitively Orienting Temporal Graphs. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 75:1-75:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{mertzios_et_al:LIPIcs.MFCS.2021.75, author = {Mertzios, George B. and Molter, Hendrik and Renken, Malte and Spirakis, Paul G. and Zschoche, Philipp}, title = {{The Complexity of Transitively Orienting Temporal Graphs}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {75:1--75:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.75}, URN = {urn:nbn:de:0030-drops-145157}, doi = {10.4230/LIPIcs.MFCS.2021.75}, annote = {Keywords: Temporal graph, transitive orientation, transitive closure, polynomial-time algorithm, NP-hardness, satisfiability} }

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**Published in:** LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

In this paper we consider the following total functional problem: Given a cubic Hamiltonian graph G and a Hamiltonian cycle C₀ of G, how can we compute a second Hamiltonian cycle C₁ ≠ C₀ of G? Cedric Smith and William Tutte proved in 1946, using a non-constructive parity argument, that such a second Hamiltonian cycle always exists. Our main result is a deterministic algorithm which computes the second Hamiltonian cycle in O(n⋅2^0.299862744n) = O(1.23103ⁿ) time and in linear space, thus improving the state of the art running time of O*(2^0.3n) = O(1.2312ⁿ) for solving this problem (among deterministic algorithms running in polynomial space). Whenever the input graph G does not contain any induced cycle C₆ on 6 vertices, the running time becomes O(n⋅ 2^0.2971925n) = O(1.22876ⁿ). Our algorithm is based on a fundamental structural property of Thomason’s lollipop algorithm, which we prove here for the first time. In the direction of approximating the length of a second cycle in a (not necessarily cubic) Hamiltonian graph G with a given Hamiltonian cycle C₀ (where we may not have guarantees on the existence of a second Hamiltonian cycle), we provide a linear-time algorithm computing a second cycle with length at least n - 4α (√n+2α)+8, where α = (Δ-2)/(δ-2) and δ,Δ are the minimum and the maximum degree of the graph, respectively. This approximation result also improves the state of the art.

Argyrios Deligkas, George B. Mertzios, Paul G. Spirakis, and Viktor Zamaraev. Exact and Approximate Algorithms for Computing a Second Hamiltonian Cycle. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 27:1-27:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{deligkas_et_al:LIPIcs.MFCS.2020.27, author = {Deligkas, Argyrios and Mertzios, George B. and Spirakis, Paul G. and Zamaraev, Viktor}, title = {{Exact and Approximate Algorithms for Computing a Second Hamiltonian Cycle}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {27:1--27:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-159-7}, ISSN = {1868-8969}, year = {2020}, volume = {170}, editor = {Esparza, Javier and Kr\'{a}l', Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.27}, URN = {urn:nbn:de:0030-drops-126953}, doi = {10.4230/LIPIcs.MFCS.2020.27}, annote = {Keywords: Hamiltonian cycle, cubic graph, exact algorithm, approximation algorithm} }

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**Published in:** LIPIcs, Volume 154, 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)

Temporal graphs are graphs whose topology is subject to discrete changes over time. Given a static underlying graph G, a temporal graph is represented by assigning a set of integer time-labels to every edge e of G, indicating the discrete time steps at which e is active. We introduce and study the complexity of a natural temporal extension of the classical graph problem Maximum Matching, taking into account the dynamic nature of temporal graphs. In our problem, Maximum Temporal Matching, we are looking for the largest possible number of time-labeled edges (simply time-edges) (e,t) such that no vertex is matched more than once within any time window of Δ consecutive time slots, where Δ ∈ ℕ is given. The requirement that a vertex cannot be matched twice in any Δ-window models some necessary "recovery" period that needs to pass for an entity (vertex) after being paired up for some activity with another entity. We prove strong computational hardness results for Maximum Temporal Matching, even for elementary cases. To cope with this computational hardness, we mainly focus on fixed-parameter algorithms with respect to natural parameters, as well as on polynomial-time approximation algorithms.

George B. Mertzios, Hendrik Molter, Rolf Niedermeier, Viktor Zamaraev, and Philipp Zschoche. Computing Maximum Matchings in Temporal Graphs. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 27:1-27:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{mertzios_et_al:LIPIcs.STACS.2020.27, author = {Mertzios, George B. and Molter, Hendrik and Niedermeier, Rolf and Zamaraev, Viktor and Zschoche, Philipp}, title = {{Computing Maximum Matchings in Temporal Graphs}}, booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)}, pages = {27:1--27:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-140-5}, ISSN = {1868-8969}, year = {2020}, volume = {154}, editor = {Paul, Christophe and Bl\"{a}ser, Markus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.27}, URN = {urn:nbn:de:0030-drops-118881}, doi = {10.4230/LIPIcs.STACS.2020.27}, annote = {Keywords: Temporal Graph, Link Stream, Temporal Line Graph, NP-hardness, APX-hardness, Approximation Algorithm, Fixed-parameter Tractability, Independent Set} }

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**Published in:** LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

Spreading processes on graphs are a natural model for a wide variety of real-world phenomena, including information or behaviour spread over social networks, biological diseases spreading over contact or trade networks, and the potential flow of goods over logistical infrastructure. Often, the networks over which these processes spread are dynamic in nature, and can be modeled with graphs whose structure is subject to discrete changes over time, i.e. with temporal graphs. Here, we consider temporal graphs in which edges are available at specified timesteps, and study the problem of deleting edges from a given temporal graph in order to reduce the number of vertices (temporally) reachable from a given starting point. This could be used to control the spread of a disease, rumour, etc. in a temporal graph. In particular, our aim is to find a temporal subgraph in which a process starting at any single vertex can be transferred to only a limited number of other vertices using a temporally-feasible path (i.e. a path, along which the times of the edge availabilities increase). We introduce a natural deletion problem for temporal graphs and we provide positive and negative results on its computational complexity, both in the traditional and the parameterised sense (subject to various natural parameters), as well as addressing the approximability of this problem.

Jessica Enright, Kitty Meeks, George B. Mertzios, and Viktor Zamaraev. Deleting Edges to Restrict the Size of an Epidemic in Temporal Networks. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 57:1-57:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{enright_et_al:LIPIcs.MFCS.2019.57, author = {Enright, Jessica and Meeks, Kitty and Mertzios, George B. and Zamaraev, Viktor}, title = {{Deleting Edges to Restrict the Size of an Epidemic in Temporal Networks}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {57:1--57:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-117-7}, ISSN = {1868-8969}, year = {2019}, volume = {138}, editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.57}, URN = {urn:nbn:de:0030-drops-110010}, doi = {10.4230/LIPIcs.MFCS.2019.57}, annote = {Keywords: Temporal networks, spreading processes, graph modification, parameterised complexity} }

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Track C: Foundations of Networks and Multi-Agent Systems: Models, Algorithms and Information Management

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

Temporal graphs are used to abstractly model real-life networks that are inherently dynamic in nature, in the sense that the network structure undergoes discrete changes over time. Given a static underlying graph G=(V,E), a temporal graph on G is a sequence of snapshots {G_t=(V,E_t) subseteq G: t in N}, one for each time step t >= 1. In this paper we study stochastic temporal graphs, i.e. stochastic processes G={G_t subseteq G: t in N} whose random variables are the snapshots of a temporal graph on G. A natural feature of stochastic temporal graphs which can be observed in various real-life scenarios is a memory effect in the appearance probabilities of particular edges; that is, the probability an edge e in E appears at time step t depends on its appearance (or absence) at the previous k steps. In this paper we study the hierarchy of models memory-k, k >= 0, which address this memory effect in an edge-centric network evolution: every edge of G has its own probability distribution for its appearance over time, independently of all other edges. Clearly, for every k >= 1, memory-(k-1) is a special case of memory-k. However, in this paper we make a clear distinction between the values k=0 ("no memory") and k >= 1 ("some memory"), as in some cases these models exhibit a fundamentally different computational behavior for these values of k, as our results indicate. For every k >= 0 we investigate the computational complexity of two naturally related, but fundamentally different, temporal path (or journey) problems: {Minimum Arrival} and {Best Policy}. In the first problem we are looking for the expected arrival time of a foremost journey between two designated vertices {s},{y}. In the second one we are looking for the expected arrival time of the best policy for actually choosing a particular {s}-{y} journey. We present a detailed investigation of the computational landscape of both problems for the different values of memory k. Among other results we prove that, surprisingly, {Minimum Arrival} is strictly harder than {Best Policy}; in fact, for k=0, {Minimum Arrival} is #P-hard while {Best Policy} is solvable in O(n^2) time.

Eleni C. Akrida, George B. Mertzios, Sotiris Nikoletseas, Christoforos Raptopoulos, Paul G. Spirakis, and Viktor Zamaraev. How Fast Can We Reach a Target Vertex in Stochastic Temporal Graphs?. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 131:1-131:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{akrida_et_al:LIPIcs.ICALP.2019.131, author = {Akrida, Eleni C. and Mertzios, George B. and Nikoletseas, Sotiris and Raptopoulos, Christoforos and Spirakis, Paul G. and Zamaraev, Viktor}, title = {{How Fast Can We Reach a Target Vertex in Stochastic Temporal Graphs?}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {131:1--131:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.131}, URN = {urn:nbn:de:0030-drops-107071}, doi = {10.4230/LIPIcs.ICALP.2019.131}, annote = {Keywords: Temporal network, stochastic temporal graph, temporal path, #P-hard problem, polynomial-time approximation scheme} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

Modern, inherently dynamic systems are usually characterized by a network structure, i.e. an underlying graph topology, which is subject to discrete changes over time. Given a static underlying graph G, a temporal graph can be represented via an assignment of a set of integer time-labels to every edge of G, indicating the discrete time steps when this edge is active. While most of the recent theoretical research on temporal graphs has focused on the notion of a temporal path and other "path-related" temporal notions, only few attempts have been made to investigate "non-path" temporal graph problems. In this paper, motivated by applications in sensor and in transportation networks, we introduce and study two natural temporal extensions of the classical problem Vertex Cover. In our first problem, Temporal Vertex Cover, the aim is to cover every edge at least once during the lifetime of the temporal graph, where an edge can only be covered by one of its endpoints at a time step when it is active. In our second, more pragmatic variation Sliding Window Temporal Vertex Cover, we are also given a natural number Delta, and our aim is to cover every edge at least once at every Delta consecutive time steps. In both cases we wish to minimize the total number of "vertex appearances" that are needed to cover the whole graph. We present a thorough investigation of the computational complexity and approximability of these two temporal covering problems. In particular, we provide strong hardness results, complemented by various approximation and exact algorithms. Some of our algorithms are polynomial-time, while others are asymptotically almost optimal under the Exponential Time Hypothesis (ETH) and other plausible complexity assumptions.

Eleni C. Akrida, George B. Mertzios, Paul G. Spirakis, and Viktor Zamaraev. Temporal Vertex Cover with a Sliding Time Window. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 148:1-148:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{akrida_et_al:LIPIcs.ICALP.2018.148, author = {Akrida, Eleni C. and Mertzios, George B. and Spirakis, Paul G. and Zamaraev, Viktor}, title = {{Temporal Vertex Cover with a Sliding Time Window}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {148:1--148:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.148}, URN = {urn:nbn:de:0030-drops-91522}, doi = {10.4230/LIPIcs.ICALP.2018.148}, annote = {Keywords: Temporal networks, temporal vertex cover, APX-hard, approximation algorithm, Exponential Time Hypothesis} }

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**Published in:** LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

In the classical binary search in a path the aim is to detect an unknown target by asking as few queries as possible, where each query reveals the direction to the target. This binary search algorithm has been recently extended by [Emamjomeh-Zadeh et al., STOC, 2016] to the problem of detecting a target in an arbitrary graph. Similarly to the classical case in the path, the algorithm of Emamjomeh-Zadeh et al. maintains a candidates’ set for the target, while each query asks an appropriately chosen vertex– the "median"–which minimises a potential \Phi among the vertices of the candidates' set. In this paper we address three open questions posed by Emamjomeh-Zadeh et al., namely (a) detecting a target when the query response is a direction to an approximately shortest path to the target, (b) detecting a target when querying a vertex that is an approximate median of the current candidates' set (instead of an exact one), and (c) detecting multiple targets, for which to the best of our knowledge no progress has been made so far. We resolve questions (a) and (b) by providing appropriate upper and lower bounds, as well as a new potential Γ that guarantees efficient target detection even by querying an approximate median each time. With respect to (c), we initiate a systematic study for detecting two targets in graphs and we identify sufficient conditions on the queries that allow for strong (linear) lower bounds and strong (polylogarithmic) upper bounds for the number of queries. All of our positive results can be derived using our new potential \Gamma that allows querying approximate medians.

Argyrios Deligkas, George B. Mertzios, and Paul G. Spirakis. Binary Search in Graphs Revisited. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 20:1-20:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{deligkas_et_al:LIPIcs.MFCS.2017.20, author = {Deligkas, Argyrios and Mertzios, George B. and Spirakis, Paul G.}, title = {{Binary Search in Graphs Revisited}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {20:1--20:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-046-0}, ISSN = {1868-8969}, year = {2017}, volume = {83}, editor = {Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.20}, URN = {urn:nbn:de:0030-drops-80589}, doi = {10.4230/LIPIcs.MFCS.2017.20}, annote = {Keywords: binary search, graph, approximate query, probabilistic algorithm, lower bound.} }

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**Published in:** LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

Finding maximum-cardinality matchings in undirected graphs is arguably one of the most central graph primitives. For m-edge and n-vertex graphs, it is well-known to be solvable in O(m\sqrt{n}) time; however, for several applications this running time is still too slow. We investigate how linear-time (and almost linear-time) data reduction (used as preprocessing) can alleviate the situation. More specifically, we focus on linear-time kernelization. We start a deeper and systematic study both for general graphs and for bipartite graphs. Our data reduction algorithms easily comply (in form of preprocessing) with every solution strategy (exact, approximate, heuristic), thus making them attractive in various settings.

George B. Mertzios, André Nichterlein, and Rolf Niedermeier. The Power of Linear-Time Data Reduction for Maximum Matching. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 46:1-46:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{mertzios_et_al:LIPIcs.MFCS.2017.46, author = {Mertzios, George B. and Nichterlein, Andr\'{e} and Niedermeier, Rolf}, title = {{The Power of Linear-Time Data Reduction for Maximum Matching}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {46:1--46:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-046-0}, ISSN = {1868-8969}, year = {2017}, volume = {83}, editor = {Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.46}, URN = {urn:nbn:de:0030-drops-81166}, doi = {10.4230/LIPIcs.MFCS.2017.46}, annote = {Keywords: Maximum-cardinality matching, bipartite graphs, linear-time algorithms, kernelization, parameterized complexity analysis, FPT in P} }

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**Published in:** LIPIcs, Volume 63, 11th International Symposium on Parameterized and Exact Computation (IPEC 2016)

Finding a vertex subset in a graph that satisfies a certain property is one of the most-studied topics in algorithmic graph theory. The focus herein is often on minimizing or maximizing the size of the solution, that is, the size of the desired vertex set. In several applications, however, we also want to limit the "exposure" of the solution to the rest of the graph. This is the case, for example, when the solution represents persons that ought to deal with sensitive information or a segregated community. In this work, we thus explore the (parameterized) complexity of finding such secluded vertex subsets for a wide variety of properties that they shall fulfill. More precisely, we study the constraint that the (open or closed) neighborhood of the solution shall be bounded by a parameter and the influence of this constraint on the complexity of minimizing separators, feedback vertex sets, F-free vertex deletion sets, dominating sets, and the maximization of independent sets.

René van Bevern, Till Fluschnik, George B. Mertzios, Hendrik Molter, Manuel Sorge, and Ondrej Suchý. Finding Secluded Places of Special Interest in Graphs. In 11th International Symposium on Parameterized and Exact Computation (IPEC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 63, pp. 5:1-5:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{vanbevern_et_al:LIPIcs.IPEC.2016.5, author = {van Bevern, Ren\'{e} and Fluschnik, Till and Mertzios, George B. and Molter, Hendrik and Sorge, Manuel and Such\'{y}, Ondrej}, title = {{Finding Secluded Places of Special Interest in Graphs}}, booktitle = {11th International Symposium on Parameterized and Exact Computation (IPEC 2016)}, pages = {5:1--5:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-023-1}, ISSN = {1868-8969}, year = {2017}, volume = {63}, editor = {Guo, Jiong and Hermelin, Danny}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2016.5}, URN = {urn:nbn:de:0030-drops-69380}, doi = {10.4230/LIPIcs.IPEC.2016.5}, annote = {Keywords: Neighborhood, Feedback Vertex Set, Vertex Deletion, Separator, Dominating Set} }

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**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

In this paper we initiate the study of populations of agents with very limited capabilities that are globally able to compute order statistics of their arithmetic input values via pair-wise meetings.
To this extent, we introduce the Arithmetic Population Protocol (APP) model, embarking from the well known Population Protocol (PP) model and inspired by two recent papers in which states are treated as integer numbers. In the APP model, every agent has a state from a set Q of states, as well as a fixed number of registers (independent of the size of the population), each of which can store an element from a totally ordered set S of samples. Whenever two agents interact with each other, they update their states and the values stored in their registers according to a joint transition function. This transition function is also restricted; it only allows (a) comparisons and (b) copy / paste operations for the sample values that are stored in the registers of the two interacting agents.
Agents can only meet in pairs via a fair scheduler and are required to eventually converge to the same output value of the function that the protocol globally and stably computes.
We present two different APPs for stably computing the median of the input values, initially stored on the agents of the population.
Our first APP, in which every agent has 3 registers and no states, stably computes (with probability 1)
the median under any fair scheduler in any strongly connected directed (or connected undirected) interaction graph.
Under the probabilistic scheduler, we show that our protocol stably computes the median in O(n^6) number of interactions in a connected undirected interaction graph of n agents.
Our second APP, in which every agent has 2 registers and O(n^2 log{n}) states, computes to the correct median of the input with high probability in O(n^3 log{n}) interactions, assuming the probabilistic scheduler and the complete interaction graph. Finally we present a third APP which, for any k, stably computes the k-th smallest element of the input of the population under any fair scheduler and in any strongly connected directed (or connected undirected) interaction graph. In this APP every agent has 2 registers and n states. Upon convergence every agent has a different state; all these states provide a total ordering of the agents with respect to their input values.

George B. Mertzios, Sotiris E. Nikoletseas, Christoforos L. Raptopoulos, and Paul G. Spirakis. Stably Computing Order Statistics with Arithmetic Population Protocols. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 68:1-68:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{mertzios_et_al:LIPIcs.MFCS.2016.68, author = {Mertzios, George B. and Nikoletseas, Sotiris E. and Raptopoulos, Christoforos L. and Spirakis, Paul G.}, title = {{Stably Computing Order Statistics with Arithmetic Population Protocols}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {68:1--68:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.68}, URN = {urn:nbn:de:0030-drops-64805}, doi = {10.4230/LIPIcs.MFCS.2016.68}, annote = {Keywords: arithmetic population protocols, order statistics, median, k-minimum element} }

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**Published in:** LIPIcs, Volume 43, 10th International Symposium on Parameterized and Exact Computation (IPEC 2015)

We study the design of fixed-parameter algorithms for problems already known to be solvable in polynomial time.
The main motivation is to get more efficient algorithms for problems with unattractive polynomial running times. Here, we focus on a fundamental graph problem: Longest Path; it is NP-hard in general but known to be solvable in O(n^4) time on n-vertex interval graphs. We show how to solve Longest Path on Interval Graphs, parameterized by vertex deletion number k to proper interval graphs, in O(k^9n) time. Notably, Longest Path is trivially solvable in linear time on proper interval graphs, and the parameter value k can be approximated up to a factor of 4 in linear time. From a more general perspective, we believe that using parameterized complexity analysis for polynomial-time solvable problems offers a very fertile ground for future studies for all sorts of algorithmic problems. It may enable a refined understanding of efficiency aspects for polynomial-time solvable problems, similarly to what classical parameterized complexity analysis does for NP-hard problems.

Archontia C. Giannopoulou, George B. Mertzios, and Rolf Niedermeier. Polynomial Fixed-parameter Algorithms: A Case Study for Longest Path on Interval Graphs. In 10th International Symposium on Parameterized and Exact Computation (IPEC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 43, pp. 102-113, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{giannopoulou_et_al:LIPIcs.IPEC.2015.102, author = {Giannopoulou, Archontia C. and Mertzios, George B. and Niedermeier, Rolf}, title = {{Polynomial Fixed-parameter Algorithms: A Case Study for Longest Path on Interval Graphs}}, booktitle = {10th International Symposium on Parameterized and Exact Computation (IPEC 2015)}, pages = {102--113}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-92-7}, ISSN = {1868-8969}, year = {2015}, volume = {43}, editor = {Husfeldt, Thore and Kanj, Iyad}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2015.102}, URN = {urn:nbn:de:0030-drops-55750}, doi = {10.4230/LIPIcs.IPEC.2015.102}, annote = {Keywords: fixed-parameter algorithm, preprocessing, data reduction, polynomial-time algorithm, longest path problem, interval graphs, proper interval vertex del} }

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**Published in:** LIPIcs, Volume 30, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)

Tolerance graphs model interval relations in such a way that intervals can tolerate a certain amount of overlap without being in conflict.
In one of the most natural generalizations of tolerance graphs with direct applications in the comparison of DNA sequences from different organisms, namely multitolerance graphs, two tolerances are allowed
for each interval - one from the left and one from the right side.
Several efficient algorithms for optimization problems that are NP-hard in general graphs have been designed for tolerance and multitolerance graphs. In spite of this progress, the complexity status of some fundamental algorithmic problems on tolerance and multitolerance graphs, such as the dominating set problem, remained unresolved until now,
three decades after the introduction of tolerance graphs. In this article we introduce two new geometric representations for tolerance and multitolerance graphs, given by points and line segments in the plane.
Apart from being important on their own, these new representations prove to be a powerful tool for deriving both hardness results and polynomial time algorithms. Using them, we surprisingly prove that the dominating set problem can be solved in polynomial time on tolerance graphs and that it is APX-hard on multitolerance graphs, solving thus a longstanding open problem. This problem is the first one that has been discovered with a different complexity status in these two graph classes. Furthermore we present an algorithm that solves the independent dominating set problem on multitolerance graphs in polynomial time,
thus demonstrating the potential of this new representation for further exploitation via sweep line algorithms.

Archontia C. Giannopoulou and George B. Mertzios. New Geometric Representations and Domination Problems on Tolerance and Multitolerance Graphs. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 354-366, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{giannopoulou_et_al:LIPIcs.STACS.2015.354, author = {Giannopoulou, Archontia C. and Mertzios, George B.}, title = {{New Geometric Representations and Domination Problems on Tolerance and Multitolerance Graphs}}, booktitle = {32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)}, pages = {354--366}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-78-1}, ISSN = {1868-8969}, year = {2015}, volume = {30}, editor = {Mayr, Ernst W. and Ollinger, Nicolas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.354}, URN = {urn:nbn:de:0030-drops-49268}, doi = {10.4230/LIPIcs.STACS.2015.354}, annote = {Keywords: tolerance graphs, multitolerance graphs, geometric representation, dominating set problem, polynomial time algorithm, APX-hard} }

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**Published in:** LIPIcs, Volume 9, 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)

Trapezoid graphs are the intersection graphs of trapezoids, where every trapezoid has a pair of opposite sides lying on two parallel lines L_{1} and L_{2} of the plane. Strictly between permutation and trapezoid graphs lie the simple-triangle graphs -- also known as PI graphs (for Point-Interval) -- where the objects are triangles with one point of the triangle on L_1 and the other two points (i.e. interval) of the triangle on L_2, and the triangle graphs -- also known as PI^* graphs -- where again the objects are triangles, but now there is no restriction on which line contains one point of the triangle and which line contains the other two. The complexity status of both triangle and simple-triangle recognition problems (namely, the problems of deciding whether a given graph is a triangle or a simple-triangle graph, respectively) have been the most fundamental open problems on these classes of graphs since their introduction two decades ago. Moreover, since triangle and simple-triangle graphs lie naturally between permutation and trapezoid graphs, and since they share a very similar structure with them, it was expected that the recognition of triangle and simple-triangle graphs is polynomial, as it is also the case for permutation and trapezoid graphs. In this article we surprisingly prove that the recognition of triangle graphs is NP-complete, even in the case where the input graph is known to be a trapezoid graph.

George B. Mertzios. The Recognition of Triangle Graphs. In 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 9, pp. 591-602, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{mertzios:LIPIcs.STACS.2011.591, author = {Mertzios, George B.}, title = {{The Recognition of Triangle Graphs}}, booktitle = {28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)}, pages = {591--602}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-25-5}, ISSN = {1868-8969}, year = {2011}, volume = {9}, editor = {Schwentick, Thomas and D\"{u}rr, Christoph}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2011.591}, URN = {urn:nbn:de:0030-drops-30469}, doi = {10.4230/LIPIcs.STACS.2011.591}, annote = {Keywords: Intersection graphs, trapezoid graphs, PI graphs, PI∗ graphs, recognition problem, NP-complete} }

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**Published in:** LIPIcs, Volume 5, 27th International Symposium on Theoretical Aspects of Computer Science (2010)

Tolerance graphs model interval relations in such a way that intervals can tolerate a certain degree of overlap without being in conflict. This subclass of perfect graphs has been extensively studied, due to both its interesting structure and its numerous applications. Several efficient algorithms for optimization
problems that are NP-hard on general graphs have been designed for tolerance graphs. In spite of this, the recognition of tolerance graphs --~namely, the problem of deciding whether a given graph is a tolerance graph~-- as well as the recognition of their main subclass of bounded tolerance graphs, have been the most fundamental open problems on this class of graphs (cf.~the book on tolerance graphs~\cite{GolTol04}) since their introduction in 1982~\cite{GoMo82}.
In this article we prove that both recognition problems are NP-complete, even in the case where the input graph is a trapezoid graph. The presented results are surprising because, on the one hand, most subclasses of perfect graphs admit polynomial recognition algorithms and, on the other hand, bounded tolerance graphs were believed to be efficiently recognizable as they are a natural special case of trapezoid graphs (which can be recognized in polynomial time)
and share a very similar structure with them. For our reduction we extend the notion of an \emph{acyclic orientation} of permutation and trapezoid graphs. Our main tool is a new algorithm that uses \emph{vertex splitting} to transform a given trapezoid graph
into a permutation graph, while preserving this new acyclic orientation property. This method of vertex splitting is of independent interest; very recently, it has been proved a powerful tool also in the design of efficient recognition algorithms for other classes of graphs~\cite{MC-Trapezoid}.

George B. Mertzios, Ignasi Sau, and Shmuel Zaks. The Recognition of Tolerance and Bounded Tolerance Graphs. In 27th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 5, pp. 585-596, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{mertzios_et_al:LIPIcs.STACS.2010.2487, author = {Mertzios, George B. and Sau, Ignasi and Zaks, Shmuel}, title = {{The Recognition of Tolerance and Bounded Tolerance Graphs}}, booktitle = {27th International Symposium on Theoretical Aspects of Computer Science}, pages = {585--596}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-16-3}, ISSN = {1868-8969}, year = {2010}, volume = {5}, editor = {Marion, Jean-Yves and Schwentick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2010.2487}, URN = {urn:nbn:de:0030-drops-24876}, doi = {10.4230/LIPIcs.STACS.2010.2487}, annote = {Keywords: Tolerance graphs, bounded tolerance graphs, recognition, vertex splitting, NP-complete, trapezoid graphs, permutation graphs} }

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