Labeled Packing of Cycles and Circuits

In 2013, Duch{\^e}ne, Kheddouci, Nowakowski and Tahraoui [4, 9] introduced a labeled version of the graph packing problem. It led to the introduction of a new parameter for graphs, the k-labeled packing number $\lambda$ k. This parameter corresponds to the maximum number of labels we can assign to the vertices of the graph, such that we will be able to create a packing of k copies of the graph, while conserving the labels of the vertices. The authors intensively studied the labeled packing of cycles, and, among other results, they conjectured that for every cycle C n of order n = 2k + x, with k $\ge$ 2 and 1 $\le$ x $\le$ 2k -- 1, the value of $\lambda$ k (C n) was 2 if x was 1 and k was even, and x + 2 otherwise. In this paper, we disprove this conjecture by giving a counter example. We however prove that it gives a valid lower bound, and we give sufficient conditions for the upper bound to hold. We then give some similar results for the labeled packing of circuits

TS-Reconfiguration of Dominating Sets in circle and circular-arc graphs

We study the dominating set reconfiguration problem with the token sliding rule. It consists, given a graph G=(V,E) and two dominating sets D_s and D_t of G, in determining if there exists a sequence S= of dominating sets of G such that for any two consecutive dominating sets D_r and D_{r+1} with r<t, D_{r+1}=(D_r\ u) U v, where uv is an edge of G. In a recent paper, Bonamy et al studied this problem and raised the following questions: what is the complexity of this problem on circular arc graphs? On circle graphs? In this paper, we answer both questions by proving that the problem is polynomial on circular-arc graphs and PSPACE-complete on circle graphs.Comment: This work was supported by ANR project GrR (ANR-18-CE40-0032) and submitted to the conference WADS 202

Linear Transformations Between Dominating Sets in the TAR-Model

Given a graph $G$ and an integer $k$, a token addition and removal ({\sf TAR} for short) reconfiguration sequence between two dominating sets $D_{\sf s}$ and $D_{\sf t}$ of size at most $k$ is a sequence $S= \langle D_0 = D_{\sf s}, D_1 \ldots, D_\ell = D_{\sf t} \rangle$ of dominating sets of $G$ such that any two consecutive dominating sets differ by the addition or deletion of one vertex, and no dominating set has size bigger than $k$. We first improve a result of Haas and Seyffarth, by showing that if $k=\Gamma(G)+\alpha(G)-1$ (where $\Gamma(G)$ is the maximum size of a minimal dominating set and $\alpha(G)$ the maximum size of an independent set), then there exists a linear {\sf TAR} reconfiguration sequence between any pair of dominating sets. We then improve these results on several graph classes by showing that the same holds for $K_{\ell}$-minor free graph as long as $k \ge \Gamma(G)+O(\ell \sqrt{\log \ell})$ and for planar graphs whenever $k \ge \Gamma(G)+3$. Finally, we show that if $k=\Gamma(G)+tw(G)+1$, then there also exists a linear transformation between any pair of dominating sets.Comment: 13 pages, 6 figure

Eternal dominating sets on digraphs and orientations of graphs

We study the eternal dominating number and the m-eternal dominating number on digraphs. We generalize known results on graphs to digraphs. We also consider the problem "oriented (m-)eternal domination", consisting in finding an orientation of a graph that minimizes its eternal dominating number. We prove that computing the oriented eternal dominating number is NP-hard and characterize the graphs for which the oriented m-eternal dominating number is 2. We also study these two parameters on trees, cycles, complete graphs, complete bipartite graphs, trivially perfect graphs and different kinds of grids and products of graphs.Comment: 34 page

Domination de graphes et problèmes de reconfiguration

This object of this thesis is to study graph domination and reconfiguration problems. A dominating set of a graph is a subset of vertices such that every vertex of the graph either is in the set, or is a neighbor of at least one vertex in the set. As for reconfiguration problems, they consist in, given a source problem, determining if it is possible, and how, to go from a solution of this source problem to another by performing a sequence of elementary changes following a given rule and that maintain a solution all along. The first chapter is a general introduction, and the second chapter is a preliminary chapter. The third chapter is devoted to reconfiguration problems. We give the definitions that are essential to understand the reconfiguration problems exposed in the following chapters. These definitions are illustrated with several problems: the 15-puzzle game and its generalizations, the reconfiguration of independent sets and dominating sets in a graph, the reconfiguration of graph colorings, the reconfiguration of the satisfiability problem, as well as the reconfiguration of connected multigraphs with the same degree sequence. The fourth chapter focuses on this last problem. We give a polynomial approximation algorithm that outputs a reconfiguration sequence between two connected multigraphs that have the same degree sequence, of length at most 2.5 times the minimum length. The elementary operation consists in exchanging an extremity between two disjoint edges. The best ratio known so far was 4, and this improvement is due to the discover of a new upper bound. However, we keep the same lower bound, and we show that as long as a better lower bound is not found, the approximation ratio can not be drastically better than 2.5. The fifth chapter goes back to domination problems. We study the connectivity and diameter of the reconfiguration graph, when the elementary operation consists in the addition or removal of a vertex of the set. In other words, we search for sufficient conditions that guarantee the existence of a reconfiguration sequence between two dominating sets, and give an upper bound on the length of this sequence. In particular, we are interested in the maximum size of the dominating sets authorized in the sequence. We show that above a certain value, that depends on the independence number of the graph, the reconfiguration graph has linear diameter. We also give another upper bound, that depends on the treewidth of the graph, above which the reconfiguration graph is connected and has linear diameter. We also give two other upper bounds for planar graphs and for K_l-minor-free graphs. In the sixth chapter, we change the elementary operation, that then consists in sliding a token along an edge (a vertex is removed from the set, and one of its neighbors is added). We study the complexity of the reachability problem, which consists in determining if there exists a reconfiguration sequence between two given dominating sets. We show that the problem is PSPACE-complete in planar bipartite graphs, unit disk graphs, circle graphs and line graphs. We also give a polynomial algorithm for circular arc graphs. The seventh chapter is devoted to the eternal domination problem. We introduce a version of the problem on directed graphs. We give some bounds on the value of the eternal domination number in two versions of the problem, that generalize results on undirected graphs. We then introduce a new problem that consists in searching for the orientation of a graph that minimizes the two parameters. We show that in the first version, determining if the parameter is at most a given k is a coNP-hard problem. We also study the value of both parameters in several graph classes such as cycles, trees, complete and complete bipartite graphs, and different types of grids. We finally characterize the graphs for which the value of the second parameter is 2Cette thèse a pour objet l'étude de la domination de graphes et des problèmes de reconfiguration. Un ensemble dominant d'un graphe est un sous-ensemble de sommets tel que tous les sommets du graphe sont ou bien dans l'ensemble, ou bien sont voisins d'au moins un sommet dans l'ensemble. Quant aux problèmes de reconfiguration, ils consistent, étant donné un problème source, à étudier si il est possible, et comment, de passer d'une solution de ce problème source à une autre en effectuant une séquence de changements élémentaires suivant une règle donnée qui maintiennent une solution. Le premier chapitre est une introduction générale et le deuxième un chapitre préliminaire. Le troisième chapitre est consacré aux problèmes de reconfiguration. Nous donnons les définitions essentielles, illustrées par divers problèmes : le jeu du taquin, la reconfiguration d'ensembles indépendants et d'ensembles dominants et de colorations de graphes, la reconfiguration du problème de satisfiabilité, ainsi que la reconfiguration de multigraphes ayant la même séquence de degrés. Le quatrième chapitre se focalise sur ce dernier problème. Nous donnons un algorithme d'approximation polynomial qui renvoit une séquence de reconfiguration entre deux multigraphes connexes ayant la même séquence de degrés, de longueur au plus 2.5 fois la longueur minimale. Le meilleur ratio obtenu jusqu'ici était de 4, et cette amélioration est due à la découverte d'une nouvelle borne supérieure. Nous gardons la même borne inférieure, et montrons que tant qu'une meilleure borne inférieure ne sera pas trouvée, le ratio d'approximation ne pourra pas être drastiquement meilleur. Le cinquième chapitre se recentre sur les problèmes de domination. Nous étudions la connexité et le diamètre du graphe de reconfiguration, lorsque l'opération élémentaire consiste en l'addition ou la suppression d'un sommet de l'ensemble dominant. En d'autres termes, nous cherchons des conditions suffisantes pour qu'il existe toujours une séquence de reconfiguration entre deux ensembles dominants, et donnons une borne supérieure sur la longueur de cette séquence. En particulier, nous nous intéressons à la taille maximale des ensembles dominants autorisés dans la séquence. Nous montrons qu'au delà d'une valeur dépendant du nombre d'indépendance du graphe, le graphe de reconfiguration a un diamètre linéaire. Nous donnons également une autre borne supérieure, qui dépend cette fois ci de la largeur arborescente du graphe, à partir de laquelle le graphe de reconfiguration est connexe et a un diamètre linéaire. Nous donnons également deux autres bornes supérieures pour les graphes planaires, ainsi que les graphes dont le graphe complet K_l n'est pas un mineur. Dans le sixième chapitre, l'opération élémentaire consiste en un glissement de jeton le long d'une arête. Nous étudions la complexité du problème d'atteignabilité, qui consiste à déterminer si il existe une séquence de reconfiguration entre deux ensembles dominants donnés. Nous montrons que le problème est PSPACE-complet pour les graphes planaires bipartis, pour les graphes de disques unité, les circle graphs, et les line graphs. Nous donnons également un algorithme polynomial pour les circular arc graphs. Le septième chapitre est consacré au problème de domination éternelle. Nous introduisons une version du problème sur les graphes dirigés. Nous donnons des bornes sur le nombre de domination éternel, dans deux versions du problèmes. Nous introduisons ensuite un problème qui consiste à chercher l'orientation d'un graphe qui minimise ces paramètres. Nous montrons que dans la première version, déterminer si le paramètre correspondant est au plus un k donné est un problème CO-NP-difficile. Nous étudions la valeur des deux paramètres sur différentes classes de graphes comme les cycles, les arbres, les graphes complets et complets bipartis, et différents types de grilles. Nous caractérisons enfin les graphes dont la valeur du deuxième paramètre est

Approximating Shortest Connected Graph Transformation for Trees

International audienceLet G, H be two connected graphs with the same degree sequence. The aim of this paper is to find a transformation from G to H via a sequence of flips maintaining connectivity. A flip of G is an operation consisting in replacing two existing edges uv, xy of G by ux and vy. Taylor showed that there always exists a sequence of flips that transforms G into H maintaining connec-tivity. Bousquet and Mary proved that there exists a 4-approximation algorithm of a shortest transformation. In this paper, we show that there exists a 2.5-approximation algorithm running in polynomial time. We also discuss the tightness of the lower bound and show that, in order to drastically improve the approximation ratio, we need to improve the best known lower bounds

TS-Reconfiguration of Dominating Sets in Circle and Circular-Arc Graphs

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