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RANDOM

**Published in:** LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

A temporal graph is a graph whose edges appear only at certain points in time. In these graphs, reachability among nodes relies on paths that traverse edges in chronological order (temporal paths). Unlike standard paths, temporal paths may not be composable, thus the reachability relation is not transitive and connected components (i.e., sets of pairwise temporally connected nodes) do not form equivalence classes, a fact with far-reaching consequences.
Recently, Casteigts et al. [FOCS 2021] proposed a natural temporal analog of the seminal Erdős-Rényi random graph model, based on the same parameters n and p. The proposed model is obtained by randomly permuting the edges of an Erdős-Rényi random graph and interpreting this permutation as an ordering of presence times. Casteigts et al. showed that the well-known single threshold for connectivity in the Erdős-Rényi model fans out into multiple phase transitions for several distinct notions of reachability in the temporal setting.
The second most basic phenomenon studied by Erdős and Rényi in static (i.e., non-temporal) random graphs is the emergence of a giant connected component. However, the existence of a similar phase transition in the temporal model was left open until now. In this paper, we settle this question. We identify a sharp threshold at p = log n/n, where the size of the largest temporally connected component increases from o(n) to n-o(n) nodes. This transition occurs significantly later than in the static setting, where a giant component of size n-o(n) already exists for any p ∈ ω(1/n) (i.e., as soon as p is larger than a constant fraction of n). Interestingly, the threshold that we obtain holds for both open and closed connected components, i.e., components that allow, respectively forbid, their connecting paths to use external nodes - a distinction arising from the absence of transitivity.
We achieve these results by strengthening the tools from Casteigts et al. and developing new techniques that provide means to decouple dependencies between past and future events in temporal Erdős-Rényi graphs, which could be of general interest in future investigations of these objects.

Ruben Becker, Arnaud Casteigts, Pierluigi Crescenzi, Bojana Kodric, Malte Renken, Michael Raskin, and Viktor Zamaraev. Giant Components in Random Temporal Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 29:1-29:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{becker_et_al:LIPIcs.APPROX/RANDOM.2023.29, author = {Becker, Ruben and Casteigts, Arnaud and Crescenzi, Pierluigi and Kodric, Bojana and Renken, Malte and Raskin, Michael and Zamaraev, Viktor}, title = {{Giant Components in Random Temporal Graphs}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)}, pages = {29:1--29:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-296-9}, ISSN = {1868-8969}, year = {2023}, volume = {275}, editor = {Megow, Nicole and Smith, Adam}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.29}, URN = {urn:nbn:de:0030-drops-188542}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2023.29}, annote = {Keywords: random temporal graph, Erd\H{o}s–R\'{e}nyi random graph, sharp threshold, temporal connectivity, temporal connected component, edge-ordered graph} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

For any hereditary graph class ℱ, we construct optimal adjacency labeling schemes for the classes of subgraphs and induced subgraphs of Cartesian products of graphs in ℱ. As a consequence, we show that, if ℱ admits efficient adjacency labels (or, equivalently, small induced-universal graphs) meeting the information-theoretic minimum, then the classes of subgraphs and induced subgraphs of Cartesian products of graphs in ℱ do too. Our proof uses ideas from randomized communication complexity and hashing, and improves upon recent results of Chepoi, Labourel, and Ratel [Journal of Graph Theory, 2020].

Louis Esperet, Nathaniel Harms, and Viktor Zamaraev. Optimal Adjacency Labels for Subgraphs of Cartesian Products. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 57:1-57:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{esperet_et_al:LIPIcs.ICALP.2023.57, author = {Esperet, Louis and Harms, Nathaniel and Zamaraev, Viktor}, title = {{Optimal Adjacency Labels for Subgraphs of Cartesian Products}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {57:1--57:11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.57}, URN = {urn:nbn:de:0030-drops-181093}, doi = {10.4230/LIPIcs.ICALP.2023.57}, annote = {Keywords: Adjacency labeling schemes, Cartesian product, Hypercubes} }

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**Published in:** LIPIcs, Volume 221, 1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022)

A temporal graph G = (G_1, G_2, ..., G_T) is a graph represented by a sequence of T graphs over a common set of vertices, such that at the i-th time step only the edge set E_i is active. The temporal graph exploration problem asks for a shortest temporal walk on some temporal graph visiting every vertex. We show that temporal graphs with n vertices can be explored in O(k n^{1.5} log n) days if the underlying graph has treewidth k and in O(n^{1.75} log n) days if the underlying graph is planar. Furthermore, we show that any temporal graph whose underlying graph is a cycle with k chords can be explored in at most 6kn days. Finally, we demonstrate that there are temporal realisations of sub cubic planar graphs that cannot be explored faster than in Ω(n log n) days. All these improve best known results in the literature.

Duncan Adamson, Vladimir V. Gusev, Dmitriy Malyshev, and Viktor Zamaraev. Faster Exploration of Some Temporal Graphs. In 1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 221, pp. 5:1-5:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{adamson_et_al:LIPIcs.SAND.2022.5, author = {Adamson, Duncan and Gusev, Vladimir V. and Malyshev, Dmitriy and Zamaraev, Viktor}, title = {{Faster Exploration of Some Temporal Graphs}}, booktitle = {1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022)}, pages = {5:1--5:10}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-224-2}, ISSN = {1868-8969}, year = {2022}, volume = {221}, editor = {Aspnes, James and Michail, Othon}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2022.5}, URN = {urn:nbn:de:0030-drops-159475}, doi = {10.4230/LIPIcs.SAND.2022.5}, annote = {Keywords: Temporal Graphs, Graph Exploration} }

Document

**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-dev.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} }

Document

**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-dev.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} }

Document

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

We give deterministic distributed (1+epsilon)-approximation algorithms for Minimum Vertex Coloring and Maximum Independent Set on chordal graphs in the LOCAL model. Our coloring algorithm runs in O( (1 / epsilon) log n) rounds, and our independent set algorithm has a runtime of O( (1/epsilon) log(1/epsilon)log^* n) rounds. For coloring, existing lower bounds imply that the dependencies on 1/epsilon and log n are best possible. For independent set, we prove that Omega(1/epsilon) rounds are necessary.
Both our algorithms make use of the tree decomposition of the input chordal graph. They iteratively peel off interval subgraphs, which are identified via the tree decomposition of the input graph, thereby partitioning the vertex set into O(log n) layers. For coloring, each interval graph is colored independently, which results in various coloring conflicts between the layers. These conflicts are then resolved in a separate phase, using the particular structure of our partitioning. For independent set, only the first O(log (1/epsilon)) layers are required as they already contain a large enough independent set. We develop a (1+epsilon)-approximation maximum independent set algorithm for interval graphs, which we then apply to those layers.
This work raises the question as to how useful tree decompositions are for distributed computing.

Christian Konrad and Viktor Zamaraev. Distributed Minimum Vertex Coloring and Maximum Independent Set in Chordal Graphs. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 21:1-21:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{konrad_et_al:LIPIcs.MFCS.2019.21, author = {Konrad, Christian and Zamaraev, Viktor}, title = {{Distributed Minimum Vertex Coloring and Maximum Independent Set in Chordal Graphs}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {21:1--21:15}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.21}, URN = {urn:nbn:de:0030-drops-109651}, doi = {10.4230/LIPIcs.MFCS.2019.21}, annote = {Keywords: local model, approximation algorithms, minimum vertex coloring, maximum independent set, chordal graphs} }

Document

**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-dev.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} }

Document

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-dev.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 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

Let vc(G), fvs(G) and oct(G) denote, respectively, the size of a minimum vertex cover, minimum feedback vertex set and minimum odd cycle transversal in a graph G. One can ask, when looking for these sets in a graph, how much bigger might they be if we require that they are independent; that is, what is the price of independence? If G has a vertex cover, feedback vertex set or odd cycle transversal that is an independent set, then we let, respectively, ivc(G), ifvs(G) or ioct(G) denote the minimum size of such a set. We investigate for which graphs H the values of ivc(G), ifvs(G) and ioct(G) are bounded in terms of vc(G), fvs(G) and oct(G), respectively, when the graph G belongs to the class of H-free graphs. We find complete classifications for vertex cover and feedback vertex set and an almost complete classification for odd cycle transversal (subject to three non-equivalent open cases).

Konrad K. Dabrowski, Matthew Johnson, Giacomo Paesani, Daniël Paulusma, and Viktor Zamaraev. On the Price of Independence for Vertex Cover, Feedback Vertex Set and Odd Cycle Transversal. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 63:1-63:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{dabrowski_et_al:LIPIcs.MFCS.2018.63, author = {Dabrowski, Konrad K. and Johnson, Matthew and Paesani, Giacomo and Paulusma, Dani\"{e}l and Zamaraev, Viktor}, title = {{On the Price of Independence for Vertex Cover, Feedback Vertex Set and Odd Cycle Transversal}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {63:1--63:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-086-6}, ISSN = {1868-8969}, year = {2018}, volume = {117}, editor = {Potapov, Igor and Spirakis, Paul and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.63}, URN = {urn:nbn:de:0030-drops-96452}, doi = {10.4230/LIPIcs.MFCS.2018.63}, annote = {Keywords: vertex cover, feedback vertex set, odd cycle transversal, price of independence} }

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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-dev.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)

Clique-width is an important graph parameter due to its algorithmic and structural properties. A graph class is hereditary if it can be characterized by a (not necessarily finite) set H of forbidden induced subgraphs. We initiate a systematic study into the boundedness of clique-width of hereditary graph classes closed under complementation. First, we extend the known classification for the |H|=1 case by classifying the boundedness of clique-width for every set H of self-complementary graphs. We then completely settle the |H|=2 case. In particular, we determine one new class of (H1, complement of H1)-free graphs of bounded clique-width (as a side effect, this leaves only six classes of (H1, H2)-free graphs, for which it is not known whether their clique-width is bounded).
Once we have obtained the classification of the |H|=2 case, we research the effect of forbidding self-complementary graphs on the boundedness of clique-width. Surprisingly, we show that for a set F of self-complementary graphs on at least five vertices, the classification of the boundedness of clique-width for ({H1, complement of H1} + F)-free graphs coincides with the one for the |H|=2 case if and only if F does not include the bull (the only non-empty self-complementary graphs on fewer than five vertices are P_1 and P_4, and P_4-free graphs have clique-width at most 2).
Finally, we discuss the consequences of our results for COLOURING.

Alexandre Blanché, Konrad K. Dabrowski, Matthew Johnson, Vadim V. Lozin, Daniël Paulusma, and Viktor Zamaraev. Clique-Width for Graph Classes Closed under Complementation. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 73:1-73:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{blanche_et_al:LIPIcs.MFCS.2017.73, author = {Blanch\'{e}, Alexandre and Dabrowski, Konrad K. and Johnson, Matthew and Lozin, Vadim V. and Paulusma, Dani\"{e}l and Zamaraev, Viktor}, title = {{Clique-Width for Graph Classes Closed under Complementation}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {73:1--73:14}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.73}, URN = {urn:nbn:de:0030-drops-80756}, doi = {10.4230/LIPIcs.MFCS.2017.73}, annote = {Keywords: clique-width, self-complementary graph, forbidden induced subgraph} }

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