Graph Searching Games and Width Measures for Directed Graphs

In cops and robber games a number of cops tries to capture a robber in a graph. A variant of these games on undirected graphs characterises tree width by the least number of cops needed to win. We consider cops and robber games on digraphs and width measures (such as DAG-width, directed tree width or D-width) corresponding to them. All of them generalise tree width and the game characterising it. For the DAG-width game we prove that the problem to decide the minimal number of cops required to capture the robber (which is the same as deciding DAG-width), is PSPACE-complete, in contrast to most other similar games. We also show that the cop-monotonicity cost for directed tree width games cannot be bounded by any function. As a consequence, D-width is not bounded in directed tree width, refuting a conjecture by Safari. A large number of directed width measures generalising tree width has been proposed in the literature. However, only very little was known about the relation between them, in particular about whether classes of digraphs of bounded width in one measure have bounded width in another. In this paper we establish an almost complete order among the most prominent width measures with respect to mutual boundedness

Recontamination Helps a Lot to Hunt a Rabbit

The Hunters and Rabbit game is played on a graph G where the Hunter player shoots at k vertices in every round while the Rabbit player occupies an unknown vertex and, if it is not shot, must move to a neighbouring vertex after each round. The Rabbit player wins if it can ensure that its position is never shot. The Hunter player wins otherwise. The hunter number h(G) of a graph G is the minimum integer k such that the Hunter player has a winning strategy (i.e., allowing him to win whatever be the strategy of the Rabbit player). This game has been studied in several graph classes, in particular in bipartite graphs (grids, trees, hypercubes...), but the computational complexity of computing h(G) remains open in general graphs and even in more restricted graph classes such as trees. To progress further in this study, we propose a notion of monotonicity (a well-studied and useful property in classical pursuit-evasion games such as Graph Searching games) for the Hunters and Rabbit game imposing that, roughly, a vertex that has already been shot "must not host the rabbit anymore". This allows us to obtain new results in various graph classes. More precisely, let the monotone hunter number mh(G) of a graph G be the minimum integer k such that the Hunter player has a monotone winning strategy. We show that pw(G) ? mh(G) ? pw(G)+1 for any graph G with pathwidth pw(G), which implies that computing mh(G), or even approximating mh(G) up to an additive constant, is NP-hard. Then, we show that mh(G) can be computed in polynomial time in split graphs, interval graphs, cographs and trees. These results go through structural characterisations which allow us to relate the monotone hunter number with the pathwidth in some of these graph classes. In all cases, this allows us to specify the hunter number or to show that there may be an arbitrary gap between h and mh, i.e., that monotonicity does not help. In particular, we show that, for every k ? 3, there exists a tree T with h(T) = 2 and mh(T) = k. We conclude by proving that computing h (resp., mh) is FPT parameterised by the minimum size of a vertex cover

On the Monotonicity of Process Number

International audienceGraph searching games involve a team of searchers that aims at capturing a fugitive in a graph. These games have been widely studied for their relationships with the tree-and the path-decomposition of graphs. In order to define de-compositions for directed graphs, similar games have been proposed in directed graphs. In this paper, we consider a game that has been defined and studied in the context of routing reconfiguration problems in WDM networks. Namely, in the processing game, the fugitive is invisible, arbitrarily fast, it moves in the opposite direction of the arcs of a digraph, but only as long as it can access to a strongly connected component free of searchers. We prove that the processing game is monotone which leads to its equivalence with a new digraph decomposition

Further results on the Hunters and Rabbit game through monotonicity

Hunters and Rabbit game is played on a graph $G$ where the Hunter player shoots at $k$ vertices in every round while the Rabbit player occupies an unknown vertex and, if not shot, must move to a neighbouring vertex after each round. The Rabbit player wins if it can ensure that its position is never shot. The Hunter player wins otherwise. The hunter number $h(G)$ of a graph $G$ is the minimum integer $k$ such that the Hunter player has a winning strategy (i.e., allowing him to win whatever be the strategy of the Rabbit player). This game has been studied in several graph classes, in particular in bipartite graphs (grids, trees, hypercubes...), but the computational complexity of computing $h(G)$ remains open in general graphs and even in trees. To progress further, we propose a notion of monotonicity for the Hunters and Rabbit game imposing that, roughly, a vertex that has already been shot ``must not host the rabbit anymore''. This allows us to obtain new results in various graph classes. Let the monotone hunter number be denoted by $mh(G)$. We show that $pw(G) \leq mh(G) \leq pw(G)+1$ for any graph $G$ with pathwidth $pw(G)$, implying that computing $mh(G)$, or even approximating $mh(G)$ up to an additive constant, is NP-hard. Then, we show that $mh(G)$ can be computed in polynomial time in split graphs, interval graphs, cographs and trees. These results go through structural characterisations which allow us to relate the monotone hunter number with the pathwidth in some of these graph classes. In all cases, this allows us to specify the hunter number or to show that there may be an arbitrary gap between $h$ and $mh$, i.e., that monotonicity does not help. In particular, we show that, for every $k\geq 3$, there exists a tree $T$ with $h(T)=2$ and $mh(T)=k$. We conclude by proving that computing $h$ (resp., $mh$) is FPT parameterised by the minimum size of a vertex cover.Comment: A preliminary version appeared in MFCS 2023. Abstract shortened due to Arxiv submission requirement

A unified FPT Algorithm for Width of Partition Functions

During the last decades, several polynomial-time algorithms have been designed that decide if a graph has treewidth (resp., pathwidth, branchwidth, etc.) at most $k$, where $k$ is a fixed parameter. Amini {\it et al.} (to appear in SIAM J. Discrete Maths.) use the notions of partitioning-trees and partition functions as a generalized view of classical decompositions of graphs, namely tree-decomposition, path-decomposition, branch-decomposition, etc. In this paper, we propose a set of simple sufficient conditions on a partition function $\Phi$, that ensures the existence of a linear-time explicit algorithm deciding if a set $A$ has $\Phi$-width at most $k$ ($k$ fixed). In particular, the algorithm we propose unifies the existing algorithms for treewidth, pathwidth, linearwidth, branchwidth, carvingwidth and cutwidth. It also provides the first Fixed Parameter Tractable linear-time algorithm deciding if the $q$-branched treewidth, defined by Fomin {\it et al.} (Algorithmica 2007), of a graph is at most $k$ ($k$ and $q$ are fixed). Our decision algorithm can be turned into a constructive one by following the ideas of Bodlaender and Kloks (J. of Alg. 1996)

Non-deterministic graph searching in trees

International audienceNon-deterministic graph searching was introduced by Fomin et al. to provide a unified approach for pathwidth, treewidth, and their interpretations in terms of graph searching games. Given q ≥ 0, the q-limited search number, s q (G), of a graph G is the smallest number of searchers required to capture an invisible fugitive in G, when the searchers are allowed to know the position of the fugitive at most q times. The search parameter s 0 (G) corresponds to the pathwidth of a graph G, and s ∞ (G) to its treewidth. Determining s q (G) is NP-complete for any fixed q ≥ 0 in general graphs and s 0 (T) can be computed in linear time in trees, however the complexity of the problem on trees has been unknown for any q > 0. We introduce a new variant of graph searching called restricted non-deterministic. The corresponding parameter is denoted by rs q and is shown to be equal to the non-deterministic graph searching parameter s q for q = 0, 1, and at most twice s q for any q ≥ 2 (for any graph G). Our main result is a polynomial time algorithm that computes rs q (T) for any tree T and any q ≥ 0. This provides a 2-approximation of s q (T) for any tree T , and shows that the decision problem associated to s 1 is polynomial in the class of trees. Our proofs are based on a new decomposition technique for trees which might be of independent interest

Παρεμποδι?σεις και Αλγο?ριθμοι για Προβλη?ματα Ανι?χνευσης Γραφημα?των

Η Ανιχνευση Γραφηματων αποτελει εναν κλαδο των Διακριτων Μαθηματικων με πληθος εφαρμογων σε πολλους τομεις της Θεωρητικης Πληροφορικης. Παρουσιαζει επισης μεγαλο θεωρητικο ενδιαφερον καθως μεσω αυτης εκφραζονται πολλα σημα- ντικα συνδυαστικα προβληματα. Στα πρωτα κεφαλαια αυτης της εργασιας θα παρουσιασουμε τα κινητρα που ωθη- σαν τους ερευνητες να ασχοληθουν με την ανιχνευση γραφηματων, θα ορισουμε τυπικα τους τρεις βασικους τυπους της και θα παρουσιασουμε τις σημαντικοτερες παραλλαγες τους. Στη συνεχεια θα αναλυσουμε τις εννοιες της μονοτονιας και της συνεκτικοτητας και θα καταγραψουμε μερικα αποτελεσματα απο τη βιβλιογραφια. Το δευτερο σκελος της εργασιας αυτης ειναι η μελετη της Θεωριας Μερικων Δια- ταξεων σε κλασεις γραφηματων και πως απο αυτη προκυπτει ο χαρακτηρισμος της κλασης μεσω ενος συνολου απαγορευμενων γραφηματων, το οποιο καλειται Συνολο Παρεμποδισης της κλασης. Αφου αναφερουμε συνοπτικα τις απαραιτητες εννοιες, θα παρουσιασουμε ολα τα εως τωρα γνωστα συνολα παρεμποδισης για κλασεις γραφημα- των με φραγμενο αριθμο ανιχνευσης. Το μεγαλυτερο σε μεγεθος συνολο που θα ανα- φερθει συγκαταλεγεται στα αποτελεσματα δικης μας εργασιας, που βρισκεται ακομα υπο συγγραφη. Η ανιχνευση γραφηματων συνδεεται αρρηκτα με τις Παραμετρους Πλατους γρα- φηματος. Τα περισσοτερα αποτελεσματα της βιβλιογραφιας αφορουν τις παραμετρους αυτες καθως η ορολογια τους διευκολυνει αρκετα την αποδειξη των θεωρηματων. Στο Κεφαλαιο 6 θα ορισθουν οι σημαντικοτερες παραμετροι πλατους γραφηματος και θα παρουσιαστει ο τροπος που σχετιζονται με τους αριθμους ανιχνευσης. Το τριτο σκελος της εργασιας αυτης αφορα την Υπολογιστικη Πολυπλοκοτητα των προβληματων ανιχνευσης γραφηματων. Στο τελος της εργασιας αυτης θα κανουμε μια συνοπτικη παρουσιαση των δικων μας αποτελεσματων και των κεντρικων ιδεων που διεπουν την αποδειξη τους.Graph Searching is a field of Discrete mathematics with numerous applications in many areas of Theoretical Computer Science. It Is also of great theoretical interest as it formalizes many important combinatorial problems. We present the motivations which led researchers to graph searching, we typically define the three basic types of searching, and we introduce some of the major variants. Then we analyze the concepts of monotonicity and connectivity and record some results from the literature. Next we study the Theory of Partial Orders on graph classes and how this is associated with the characterization of some classes through a set of forbidden graphs, called obstruction Set of the class. After briefly mentioning the necessary concepts, we present all so far known obstruction sets for classes of graphs with bounded search number. The larger set to be mentioned is included in the results of our work, which is still under preparation. Graph searching is closely related to the Width Parameters of a graph. Most of the results in the literature concern these parameters, as their terminology eases the proofs of the theorems. In Chapter 6 we define some important width parameters and we illustrate how they relate to the search number of the graph. The third part of this work consists of the study of the Computational Complexity of some graph searching problems. Finally we make a brief presentation of our results and the core ideas underlying their proofs

An Unified FPT Algorithm for Width of Partition Functions

During the last decades, several polynomial-time algorithms have been designed that decide whether a graph has tree-width (resp., path-width, branch-width, etc.) at most k, where k is a fixed parameter. Amini et al. (Discrete Mathematics'09) use the notions of partitioning-trees and partition functions as a generalized view of classical decompositions of graphs, namely tree decomposition, path decomposition, branch decomposition, etc. In this paper, we propose a set of simple sufficient conditions on a partition function Φ, that ensures the existence of a linear-time explicit algorithm deciding if a set A has Φ-width at most k (k fixed). In particular, the algorithm we propose unifies the existing algorithms for tree-width, path-width, linear-width, branch-width, carving-width and cut-width. It also provides the first Fixed Parameter Tractable linear-time algorithm to decide if the q-branched tree-width, defined by Fomin et al. (Algorithmica'09), of a graph is at most k (k and q are fixed). Moreover, the algorithm is able to decide if the special tree-width, defined by Courcelle (FSTTCS'10), is at most k, in linear-time where k is a Fixed Parameter. Our decision algorithm can be turned into a constructive one by following the ideas of Bodlaender and Kloks (J. of Alg. 1996).Au cours de ces dernières années, plusieurs algorithmes polynomiaux ont été conçus pour décider si un graphe a largeur arborescente (resp., largeur en chemin, branch-width, etc) au plus k, où k est un paramètre fixe. Amini et al. (Discrete Mathematics'09) ont utilisé les notions d'arbres de partition et de fonctions de partition comme une vision généralisée des décompositions des graphes classiques, à savoir la décomposition arborescente, la décomposition en chemin, la décomposition en branche, etc. Dans cet article, nous proposons un ensemble de conditions sur une fonction de partition Φ, qui assure l'existence d'un algorithme explicite en temps linéaire pour décider si un ensemble A a Φ-largeur au plus k (oú k est fixé). En particulier, l'algorithme que nous proposons unifie les algorithmes existants pour la largeur arborescente, largeur en chemin, la largeur linéaire, la largeur de branche, cut-width et carving-width. Il est également le premier algorithme FPT pour décider si la largeur arborescente q-ramifié, définie par Fomin et al. (Algorithmica'09), d'un graphe est au plus k (k et q sont fixées). De plus, l'algorithme est capable de décider si la largeur arborescente spéciale, définie par Courcelle (FSTTCS'10), est plus k, où k est un paramètre fixé. Notre algorithme de décision peut être transformé en un algorithme constructif en suivant les idées de Bodlaender et Kloks (J. of Alg., 1996)