The complexity of the Perfect Matching-Cut problem

Perfect Matching-Cut is the problem of deciding whether a graph has a perfect matching that contains an edge-cut. We show that this problem is NP-complete for planar graphs with maximum degree four, for planar graphs with girth five, for bipartite five-regular graphs, for graphs of diameter three and for bipartite graphs of diameter four. We show that there exist polynomial time algorithms for the following classes of graphs: claw-free, $P_5$-free, diameter two, bipartite with diameter three and graphs with bounded tree-width

On the vertices belonging to all, some, none minimum dominating set

11 figuresWe characterize the vertices belonging to all minimum dominating sets, to some minimum dominating sets but not all, and to no minimum dominating set. We refine this characterization for some well studied sub-classes of graphs: chordal, claw-free, triangle-free. Also we exhibit some graphs answering to some open questions of the literature on minimum dominating sets

On the complexity of Dominating Set for graphs with fixed diameter

A set $S\subseteq V$ of a graph $G=(V,E)$ is a dominating set if each vertex has a neighbor in $S$ or belongs to $S$. Dominating Set is the problem of deciding, given a graph $G$ and an integer $k\geq 1$, if $G$ has a dominating set of size at most $k$. It is well known that this problem is $\mathsf{NP}$-complete even for claw-free graphs. We give a complexity dichotomy for Dominating Set for the class of claw-free graphs with diameter $d$. We show that the problem is $\mathsf{NP}$-complete for every fixed $d\ge 3$ and polynomial time solvable for $d\le 2$. To prove the case $d=2$, we show that Minimum Maximal Matching can be solved in polynomial time for $2K_2$-free graphs.Comment: 15 pages, 5 figure

On Minimum Dominating Sets in cubic and (claw,H)-free graphs

Given a graph $G=(V,E)$, $S\subseteq V$ is a dominating set if every $v\in V\setminus S$ is adjacent to an element of $S$. The Minimum Dominating Set problem asks for a dominating set with minimum cardinality. It is well known that its decision version is $NP$-complete even when $G$ is a claw-free graph. We give a complexity dichotomy for the Minimum Dominating Set problem for the class of $(claw, H)$-free graphs when $H$ has at most six vertices. In an intermediate step we show that the Minimum Dominating Set problem is $NP$-complete for cubic graphs

Biological activities of nitidine, a potential anti-malarial lead compound

International audienceAbstract Background Nitidine is thought to be the main active ingredient in several traditional anti-malarial remedies used in different parts of the world. The widespread use of these therapies stresses the importance of studying this molecule in the context of malaria control. However, little is known about its potential as an anti-plasmodial drug, as well as its mechanism of action. Methods In this study, the anti-malarial potential of nitidine was evaluated in vitro on CQ-sensitive and -resistant strains. The nitidine's selectivity index compared with cancerous and non-cancerous cell lines was then determined. In vivo assays were then performed, using the four-day Peter's test methodology. To gain information about nitidine's possible mode of action, its moment of action on the parasite cell cycle was studied, and its localization inside the parasite was determined using confocal microscopy. The in vitro abilities of nitidine to bind haem and to inhibit β-haematin formation were also demonstrated. Results Nitidine showed similar in vitro activity in CQ-sensitive and resistant strains, and also a satisfying selectivity index (> 10) when compared with a non-cancerous cells line. Its in vivo activity was moderate; however, no sign of acute toxicity was observed during treatment. Nitidine's moment of action on the parasite cycle showed that it could not interfere with DNA replication; this was consistent with the observation that nitidine did not localize in the nucleus, but rather in the cytoplasm of the parasite. Nitidine was able to form a 1-1 complex with haem in vitro and also inhibited β-haematin formation with the same potency as chloroquine. Conclusion Nitidine can be considered a potential anti-malarial lead compound. Its ability to complex haem and inhibit β-haematin formation suggests a mechanism of action similar to that of chloroquine. The anti-malarial activity of nitidine could therefore be improved by structural modification of this molecule to increase its penetration of the digestive vacuole in the parasite, where haemoglobin metabolization takes place

The bondage number of chordal graphs

A set $S\subseteq V(G)$ of a graph $G$ is a dominating set if each vertex has a neighbor in $S$ or belongs to $S$. Let $\gamma(G)$ be the cardinality of a minimum dominating set in $G$. The bondage number $b(G)$ of a graph $G$ is the smallest cardinality of a set edges $A\subseteq E(G)$ such that $\gamma(G-A)=\gamma(G)+1$. A chordal graph is a graph with no induced cycle of length four or more. In this paper, we prove that the bondage number of a chordal graph $G$ is at most the order of its maximum clique, that is, $b(G)\leq \omega(G)$. We show that this bound is best possible

Domination problems and partitions of graphs : complexity, structure, criticality

Cette thèse porte sur des problèmes et des questions de la théorie et de l'algorithmique de graphe. La première partie concerne l'étude de l'ensemble dominant minimum. Nous considérons son problème de décision et montrons que le nombre de domination peut être déterminé en temps polynomial dans la classe des graphes sans griffe et sans chemin induit de taille au plus huit. Ensuite, nous étudions les sommets à l'intersection de tous les ensembles dominants minimums d'un graphe. Nous montrons notamment que pour certaines classes de graphes, ces sommets sont toujours critiques relativement au nombre de domination. Nous finissons cette partie sur l'étude des arêtes critiques, textit{i.e.} les arêtes dont la suppression fait varier le nombre de domination. Nous montrons dans un premier temps une borne optimale sur la cardinalité minimum d'un ensemble d'arêtes critiques dans les graphes triangulés. Dans un second temps, nous montrons qu'il est difficile de décider si il existe une arête critique dans des sous classes des graphes planaires. La seconde partie porte sur des problèmes de partitionnement des sommets du graphe. 'Etant donné une partition des sommets, nous introduisons une mesure de satisfaction pour chaque sommet, qui est le rapport entre son nombre de voisins dans sa communauté et le nombre total de ses voisins. Nous posons ensuite le problème suivant: quel est le ratio maximum a/b tel qu'un graphe possède une partition o`u chaque sommet à une satisfaction d'au moins a/b ? Nous étudions les valeurs minimums que ce critère de satisfaction peut atteindre dans différentes classes de graphes. Nous montrons aussi que ce problème est lié au problème du couplage-disconnectant dans les graphes réguliers. Pour cette raison nous poursuivons cette thèse sur l'étude du problème du couplage parfait-disconnectant. Celui-ci consiste à décider si un graphe possède une couplage parfait qui contient une coupe. Nous montrons la complexité de ce problème, notamment dans des classes de graphes planaires; de degré fixé; bipartis; de diamètre au plus d; sans-griffe; sans-P_5.This thesis relates to the study of some problems and questions arising from graph theory and algorithmic theory. The first part is devoted to the study of the minimum dominating set. We consider its decision problem and show that the domination number is computable in polynomial time for graphs with no induced claw and no induced path of length at most height. Then, we focus on vertices at the intersection of every minimum dominating sets of a graph. We show that for certain classes of graphs, these vertices are always critical relatively to the domination number. The first part ends with the study of critical edges, that is, edges which removal increase the domination number. We show first an optimal bound on the minimum cardinality of a critical set of edges for chordal graphs. Then, we show that it is difficult to decide if there is a critical edge in subclasses of planar graphs. The last part deals with partitions of the vertices of the graph. Given a graph and a partition of its vertices, we introduce a satisfaction criteria for each vertex, which is the ratio between the number of its neighbors in its community and the total number of its neighbors. We give the following problem: what is the maximum ratio a/b such that a graph has a partition where each vertex has a satisfaction of at least a/b? We study lower bounds of this optimal satisfaction criteria for some classes of graphs. We also show that this problem is related to the matching-cut problem in regular graphs. For this reason, we pursue this thesis with the study of the perfect matching-cut problem. It consists in deciding if a graph has a perfect matching that contains a cut. We give complexity results, notably in graph classes that are planar; with fixed degree; that are bipartite; with a fixed diameter d; that are claw-free; that are P5-free

The complexity of the Bondage problem in planar graphs

20 pages, 12 figuresA set $S\subseteq V(G)$ of a graph $G$ is a dominating set if each vertex has a neighbor in $S$ or belongs to $S$. Let $\gamma(G)$ be the cardinality of a minimum dominating set in $G$. The bondage number $b(G)$ of a graph $G$ is the smallest number of edges $A\subseteq E(G)$, such that $\gamma(G-A)=\gamma(G)+1$. The problem of finding $b(G)$ for a graph $G$ is known to be NP-hard even for bipartite graphs. In this paper, we show that deciding if $b(G)=1$ is NP-hard, while deciding if $b(G)=2$ is coNP-hard, even when $G$ is restricted to one of the following classes: planar $3$-regular graphs, planar claw-free graphs with maximum degree $3$, planar bipartite graphs of maximum degree $3$ with girth $k$, for any fixed $k\geq 3$