Search

International audience; The node search game against a lazy (or, respectively, agile) invisible robber has been introduced as a search-game analogue of the treewidth parameter (and, respectively, pathwidth). In the \emph{connected} variants of the above two games, we additionally demand that, at each moment of the search, the \emph{clean} territories are {\sl connected}. The connected search game against an agile and invisible robberhas been extensively examined. The monotone variant (where we also demand that the clean territories are progressively increasing) of this game, corresponds to the graph parameter of {\sl connected pathwidth}. It is known that \emph{the price of connectivtiy} to search for an agile robber is bounded by $2$, that is the connected pathwidth of a graph is at most twice (plus some constant) its pathwidth.In this paper, we investigate the study of the connected search game against a {\sl lazy} robber. A lazy robber moves only when the searchers' strategy threatens the location that he currently occupies. We introduce two alternative graph-theoretical formulations of this game, one in terms of connected tree decompositions, and one in terms of (connected) layouts, leading to the graph parameter of {\sl connected treewidth}. We observe that connected treewidth parameter is closed under contractions and prove that for every $k\geq 2$, the set of contraction obstructions of the class of graphs with connected treewidth at most $k$ is infinite.Our main result is a complete characterisation of the obstruction set for $k=2$. One may observe that, so far, only a few complete obstruction sets are explicity known for contraction closed graph classes.We finally show that, in contrast to the agile robber game, the price of connectivity is unbounded.

Research on the similarity of a graph to being a tree—called the treewidth of the graph—has seen an enormous rise within the last decade, but a practically fast algorithm for this task has been discovered only recently by Tamaki (ESA 2017). It is based on dynamic programming and makes use of the fact that the number of positive subinstances is typically substantially smaller than the number of all subinstances. Algorithms producing only such subinstances are called positive-instance driven (PID). The parameter treedepth has a similar story. It was popularized through the graph sparsity project and is theoretically well understood—but the first practical algorithm was discovered only recently by Trimble (IPEC 2020) and is based on the same paradigm. We give an alternative and unifying view on such algorithms from the perspective of the corresponding configuration graphs in certain two-player games. This results in a single algorithm that can compute a wide range of important graph parameters such as treewidth, pathwidth, and treedepth. We complement this algorithm with a novel randomized data structure that accelerates the enumeration of subproblems in positive-instance driven algorithms.

International audience; The graph parameter of {\sl pathwidth} can be seen as a measure of the topological resemblance of a graph to a path. A popular definition of pathwidth is given in terms of {\sl node search} where we are given a system of tunnels (represented by a graph) that is contaminated by some infectious substance and we are looking for a search strategy that, at each step, either places a searcher on a vertex or removes a searcher from a vertex and where an edge is cleaned when both endpoints are simultaneously occupied by searchers. It was proved that the minimum number of searchers required for a successful cleaning strategy is equal to the pathwidth of the graph plus one.Two desired characteristics for a cleaning strategy is to be {\sl monotone} (no recontamination occurs) and {\sl connected} (clean territories always remain connected). Under these two demands, the number of searchers is equivalent to a variant of pathwidth called {\em connected pathwidth}. We prove that connected pathwidth is fixed parameter tractable, in particular we design a $2^{O(k^2)}\cdot n$ time algorithm that checks whether the connected pathwidth of $G$ is at most $k.$ This resolves an open question by [{\sl Dereniowski, Osula, and Rz{\k{a}}{\.{z}}ewski, Finding small-width connected path-decompositions in polynomial time. Theor. Comput. Sci., 794:85–100, 2019}\,]. For our algorithm, we enrich the {\sl typical sequence technique} that is able to deal with the connectivity demand. Typical sequences have been introduced in [{\sl Bodlaender and Kloks. Efficient and constructive algorithms for the pathwidth and treewidth of graphs. J. Algorithms, 21(2):358–402, 1996}\,] for the design of linear parameterized algorithms for treewidth and pathwidth. While this technique has been later applied to other parameters, none of its advancements was able to deal with the connectivity demand, as it is a ``global’’ demand that concerns an unbounded number of parts of the graph of unbounded size. The proposed extension is based on an encoding of the connectivity property that is quite versatile and may be adapted so to deliver linear parameterized algorithms for the connected variants of other width parameters as well. An immediate consequence of our result is a $2^{O(k^2)}\cdot n$ time algorithm for the monotone and connected version of the edge search number.

International audience; In Graph Searching, a team of searchers aims at capturing an invisible fugitive moving arbitrarily fast in a graph. Equivalently, the searchers try to clear a contaminated network. The problem is to compute the minimum number of searchers required to accomplish this task. Several variants of Graph Searching have been studied mainly because of their close relationship with the pathwidth of a graph. Blin et al. defined the Exclusive Graph Searching where searchers cannot " jump " and no node can be occupied by more than one searcher. In this paper, we study the complexity of this new variant. We show that the problem is NP-hard in planar graphs with maximum degree 3 and it can be solved in linear-time in the class of cographs. We also show that monotone Exclusive Graph Searching is NP-complete in split graphs where Pathwidth is known to be solvable in polynomial time. Moreover, we prove that monotone Exclusive Graph Searching is in P in a subclass of star-like graphs where Pathwidth is known to be NP-hard. Hence, the computational complexities of monotone Exclusive Graph Searching and Pathwidth cannot be compared. This is the first variant of Graph Searching for which such a difference is proved.

International audience; Non-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.

In the edge searching problem searchers move from vertex to vertex in a graph to capture an invisible fast intruder that may occupy either vertices or edges. Fast searching is a monotonic internal model in which at every move a new edge of the graph G must be guaranteed to be free of the intruder. That is once all searchers are placed the graph G is cleared in exactly vertical bar E(G)vertical bar moves. Such a restriction obviously necessitates a larger number of searchers. We examine this model and characterize graphs for which 2 or 3 searchers are sufficient. We prove that the corresponding decision problem is NP-complete. (C) 2013 Elsevier B.V. All rights reserved.

The problems of (classical) searching and connected searching of weighted trees are known to be computationally hard. In this work we give a polynomial-time 3-approximation algorithm that finds a connected search strategy of a given weighted tree. This in particular yields constant factor approximation algorithms for the (non-connected) classical searching problems and for the weighted pathwidth problem for this class of graphs.

International audience; We introduce a variant of the classic node search game called LIFO-search where searchers are assigned different numbers. The additional rule is that a searcher can be removed only if no searchers of lower rank are in the graph at that moment. We show that all common variations of the game require the same number of searchers. We then introduce the notion of (directed) shelters in (di)graphs and prove a min-max theorem implying their equivalence to the cycle-rank/tree-depth parameter in (di)graphs. As (directed) shelters provide escape strategies for the fugitive, this implies that the LIFO-search game is monotone and that the LIFO-search parameter is equivalent to the one of cycle-rank/tree-depth in (di)graphs.

Graph 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 tree- and path-decomposition of graphs. In order to define decompositions for directed graphs, similar games have been proposed in directed graphs. In this paper, we consider such 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, arbitrary fast, it moves in the opposite direction of the arcs of a digraph, but only as long as it has 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.; Les jeux de capture impliquent une équipe d'agents qui doivent capturer un fugitif se déplaçant dans un graphe. Ces jeux ont été très étudiés pour leur interprétation en termes de décompositions de graphes (tree-decomposition, path-decomposition). Dans le but de définir de "bonnes" décompositions pour les graphes orientés, des jeux similaires ont été définis et étudiés dans les graphes orientés. Dans ce travail, nous considérons un jeu qui a été défini dans le contexte du routage dans les réseaux WDM. Dans ce jeu, le processing game, le fugitif est invisible, arbitrairement rapide et se déplace dans le sens opposé des arcs. De plus, le fugitif est capturé dès qu'il lui est impossible d'accéder à une composante fortement connexe sans agents. Nous prouvons que ce jeu est monotone. Cela permet de montrer l'équivalence du processing game et d'une nouvelle décomposition de graphes orientés.

We give a linear-time algorithm for computing the edge search number of cographs, thereby resolving the computational complexity of edge searching on this graph class. To achieve this we give a characterization of the edge search number of the join of two graphs. With our result, the knowledge on graph searching of cographs is now complete: node, mixed, and edge search numbers of cographs can all be computed efficiently. Furthermore, we are one step closer to computing the edge search number of permutation graphs.

In Graph Searching, a team of searchers aims at capturing an invisible fugitive moving arbitrarily fast in a graph. Equivalently, the searchers try to clear a contaminated network.The problem is to compute the minimum number of searchers required to accomplish this task. Several variants of Graph Searching have been studied mainly because of their close relationship with the pathwidth of a graph. Blin et al. defined the Exclusive Graph Searching where searchers cannot "jump" and no node can be occupied by more than one searcher. In this paper, we study the complexity of this new variant. We show that the problem is NP-hard in planar graphs with maximum degree 3 and it can be solved in linear-time in the class of cographs. We also show that monotone Exclusive Graph Searching is NP-complete in split graphs where Pathwidth is known to be solvable in polynomial time. Moreover, we prove that monotone Exclusive Graph Searching is in P in a subclass of star-like graphs where Pathwidth is known to be NP-hard. Hence, the computational complexities of monotone Exclusive Graph Searching and Pathwidth cannot be compared. This is the first variant of Graph Searching for which such a difference is proved.; Dans les jeux de capture (Graph Searching), une équipe d'agents doit capturer un fugitif invisible se déplaçant rapidement dans un graphe. De façon équivalente, les agents doivent nettoyer un réseau contaminé. Le problème est de calculer le nombre minimum d'agents nécessaires pour accomplir cette tache. Plusieurs variantes de ces jeux ont été étudiés pour leur lien avec la pathwidth des graphes. Blin et al. ont défini le jeu de capture exclusif dans lequel les agents ne peuvent ni "sauter", ni occuper un sommet à plusieurs. Dans ce rapport, nous étudions la complexité de cette nouvelle variante. Nous montrons que le problème est NP-difficile dans les graphes planaires avec degré au plus 3 et qu'il peut être résolu en temps linéaire dans la classe des cographes. Nous montrons également que la variante monotone du jeu exclusif est NP-complet dans la classe des split-graphes où la pathwidth est NP-complet. De plus, nous prouvons que le jeu exclusif monotone peut être calculé en temps polynomial dans une classe de star-like graphes où calculer la pathwidth est NP-complet. Les complexités de la pathwidth et du jeu de capture exclusif monotone ne peuvent donc être comparées. Il s'agit de la première variante de jeu de capture où une telle différence est avérée

The linear-width of a graph G is defined to be the smallest integer k such that the edges of G can be arranged in a linear ordering (e1,…,er) in such a way that for every i=1,…,r−1, there are at most k vertices incident to edges that belong both to {e1,…,ei} and to {ei+1,…,er}. In this paper, we give a set of 57 graphs and prove that it is the set of the minimal forbidden minors for the class of graphs with linear-width at most two. Our proof also gives a linear time algorithm that either reports that a given graph has linear-width more than two or outputs an edge ordering of minimum linear-width. We further prove a structural connection between linear-width and the mixed search number which enables us to determine, for any k⩾1, the set of acyclic forbidden minors for the class of graphs with linear-width⩽k. Moreover, due to this connection, our algorithm can be transfered to two linear time algorithms that check whether a graph has mixed search or edge search number at most two and, if so, construct the corresponding sequences of search moves.

Non-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_{\infty}(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 that we call restricted non-deterministic. The corresponding parameter is denoted by rs_q and is shown to be equal to s_q for q=0,1, and at most twice s_q for any q>=2 (for any graph G). Our main result is the design of 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. We also prove that the number of queries required to search a tree with two searchers can be computed in linear time. Tight upper bounds on the minimum number of queries for an arbitrary fixed number of searchers are also provided.

International audience; We present a distributed algorithm to compute the node search number in trees. This algorithm extends the centralized algorithm proposed by Ellis \emph{et al.}~[EST94]. It can be executed in an asynchronous environment, requires an overall computation time of $O(n\log{n})$, and $n$ messages of $\log_3{n}+4$ bits each. The main contribution of this work lies in the data structure proposed to design our algorithm, called \emph{hierarchical decomposition}. This simple and flexible data structure is used for four operations: updating the node search number after addition or deletion of any tree-edges in a distributed fashion; computing it in a tree whose edges are added sequentially and in any order; computing other graph invariants such as the process number and the edge search number, by changing only initialization rules; extending our algorithms for trees and forests of unknown size (using messages of up to $2\log_3{n}+5$ bits).

It is proven that the connected pathwidth of any graph $G$ is at most $2\cdot\textup{pw}(G)+1$, where $\textup{pw}(G)$ is the pathwidth of $G$. The method is constructive, i.e., it yields an efficient algorithm that for a given path decomposition of width $k$ computes a connected path decomposition of width at most $2k+1$. The running time of the algorithm is $O(dk^2)$, where $d$ is the number of “bags” in the input path decomposition. The motivation for studying connected path decompositions comes from the connection between the pathwidth and the search number of a graph. One of the advantages of the above bound for connected pathwidth is an inequality $\textup{\texttt{cs}}(G)\leq 2\textup{\texttt{s}}(G)+3$, where $\textup{\texttt{cs}}(G)$ and $\textup{\texttt{s}}(G)$ are the connected search number and the search number of $G$, respectively. Moreover, the algorithm presented in this work can be used to convert a given search strategy using $k$ searchers into a (monotone) connected one using $2k+3$ searc...

International audience; We introduce nondeterministic graph searching with a controlled amount of nondeterminism and show how this new tool can be used in algorithm design and combinatorial analysis applying to both pathwidth and treewidth. We prove equivalence between this game-theoretic approach and graph decompositions called q -branched tree decompositions, which can be interpreted as a parameterized version of tree decompositions. Path decomposition and (standard) tree decomposition are two extreme cases of q-branched tree decompositions. The equivalence between nondeterministic graph searching and q-branched tree decomposition enables us to design an exact (exponential time) algorithm computing q-branched treewidth for all q≥0, which is thus valid for both treewidth and pathwidth. This algorithm performs as fast as the best known exact algorithm for pathwidth. Conversely, this equivalence also enables us to design a lower bound on the amount of nondeterminism required to search a graph with the minimum number of searchers.

International audience; Search games are attractive for their correspondence with classical width parameters. For instance, the \emph{invisible} search number (a.k.a. \emph{node} search number) of a graph is equal to its pathwidth plus~1, and the \emph{visible} search number of a graph is equal to its treewidth plus~1. The \emph{connected} variants of these games ask for search strategies that are connected, i.e., at every step of the strategy, the searched part of the graph induces a connected subgraph. We focus on \emph{monotone} search strategies, i.e., strategies for which every node is searched exactly once. The monotone connected visible search number of an $n$-node graph is at most $O(\log n)$ times its visible search number. First, we prove that this logarithmic bound is tight. Precisely, we prove that there is an infinite family of graphs for which the ratio monotone connected visible search number over visible search number is $\Omega(\log n)$. Second, we prove that, as opposed to the non-connected variant of visible graph searching, ``recontamination helps" for connected visible search. Precisely, we prove that, for any $k \geq 4$, there exists a graph with connected visible search number at most $k$, and monotone connected visible search number $>k$.

Expansion is the way of generalizing different graph layout and searching problems. We initiate the study of connected expansion which naturally arises in a number of applications. Our main tool for this investigation is the branchwidth of a graph. In particular, we prove that any 2-edge-connected graph of branchwidth k has a {em connected} branch decom-po-si-tion of width k, i.e., a branch decomposition in which any cut separates two edge-sets that induce two connected subgraphs. Our proof is constructive, and is inspired from the existential proof of Seymour and Thomas (1994) for carvings. We also prove that the {em connected} search number (i.e., connected pathwidth) of any n-node graph of branchwidth k is at most O(klog n) and this bound is the best possible for parameters k and n. A first consequence of these results is that, for any graph, the connected search number is at most O(log n) times larger than the (standard) search number. The only bound known so far held for trees only. Another consequence is that the connected search number can be approximated in polynomial time up to a factor O(log{n}cdotlog{OPT}). That is, for any connected graph G, one can compute in polynomial time a connected search strategy for G that uses at most O (log{n}cdotlog{OPT}) times the optimal number OPT of searchers. The ratio O(log{n}cdotlog{OPT}) is the same as the best known approximation ratio for (standard) pathwidth, and any improvement for the approximation of connected search (i.e., connected pathwidth) would indeed produce an improvement for the approximation of pathwidth.