On the hull number of triangle-free graphs

A set of vertices C in a graph is convex if it contains all vertices which lie on shortest paths between vertices in C. The convex hull of a set of vertices S is the smallest convex set containing S. The hull number h(G) of a graph G is the smallest cardinality of a set of vertices whose convex hull is the vertex set of G. For a connected triangle-free graph G of order n and diameter d\geq 3, we prove that h(G)\leq (n-d+3)/3, if G has minimum degree at least 3 and that h(G)\leq 2(n-d+5)/7, if G is cubic. Furthermore, for a connected graph G of order n, girth g\geq 4, minimum degree at least 2, and diameter d, we prove h(G)\leq 2+ (n-d-1)/\left\lceil\frac{g-1}{2}\right \rceil. All bounds are best possible

The Geodetic Hull Number is Hard for Chordal Graphs

We show the hardness of the geodetic hull number for chordal graphs

Convexity in partial cubes: the hull number

We prove that the combinatorial optimization problem of determining the hull number of a partial cube is NP-complete. This makes partial cubes the minimal graph class for which NP-completeness of this problem is known and improves some earlier results in the literature. On the other hand we provide a polynomial-time algorithm to determine the hull number of planar partial cube quadrangulations. Instances of the hull number problem for partial cubes described include poset dimension and hitting sets for interiors of curves in the plane. To obtain the above results, we investigate convexity in partial cubes and characterize these graphs in terms of their lattice of convex subgraphs, improving a theorem of Handa. Furthermore we provide a topological representation theorem for planar partial cubes, generalizing a result of Fukuda and Handa about rank three oriented matroids.Comment: 19 pages, 4 figure

Computing metric hulls in graphs

We prove that, given a closure function the smallest preimage of a closed set can be calculated in polynomial time in the number of closed sets. This confirms a conjecture of Albenque and Knauer and implies that there is a polynomial time algorithm to compute the convex hull-number of a graph, when all its convex subgraphs are given as input. We then show that computing if the smallest preimage of a closed set is logarithmic in the size of the ground set is LOGSNP-complete if only the ground set is given. A special instance of this problem is computing the dimension of a poset given its linear extension graph, that was conjectured to be in P. The intent to show that the latter problem is LOGSNP-complete leads to several interesting questions and to the definition of the isometric hull, i.e., a smallest isometric subgraph containing a given set of vertices $S$. While for $|S|=2$ an isometric hull is just a shortest path, we show that computing the isometric hull of a set of vertices is NP-complete even if $|S|=3$. Finally, we consider the problem of computing the isometric hull-number of a graph and show that computing it is $\Sigma^P_2$ complete.Comment: 13 pages, 3 figure

Hull number: P5-free graphs and reduction rules

International audienceIn this paper, we study the (geodesic) hull number of graphs. For any two vertices $u,v\in V$ of a connected undirected graph $G=(V,E)$, the closed interval $I[u,v]$ of $u$ and $v$ is the set of vertices that belong to some shortest $(u,v)$-path. For any $S \subseteq V$, let $I[S]= \bigcup_{u,v\in S} I[u,v]$. A subset $S\subseteq V$ is (geodesically) convex if $I[S] = S$. Given a subset $S\subseteq V$, the convex hull $I_h[S]$ of $S$ is the smallest convex set that contains $S$. We say that $S$ is a hull set of $G$ if $I_h[S] = V$. The size of a minimum hull set of $G$ is the hull number of $G$, denoted by $hn(G)$. First, we show a polynomial-time algorithm to compute the hull number of any $P_5$-free triangle-free graph. Then, we present four reduction rules based on vertices with the same neighborhood. We use these reduction rules to propose a fixed parameter tractable algorithm to compute the hull number of any graph $G$, where the parameter can be the size of a vertex cover of $G$ or, more generally, its neighborhood diversity, and we also use these reductions to characterize the hull number of the lexicographic product of any two graphs

On the hull number of some graph classes

In this paper, we study the geodetic convexity of graphs focusing on the problem of the complexity to compute inclusion-minimum hull set of a graph in several graph classes. For any two vertices $u,v\in V$ of a connected graph $G=(V,E)$, the {\em closed interval} $I[u,v]$ of $u$ and $v$ is the the set of vertices that belong to some shortest $(u,v)$-path. For any $S \subseteq V$, let $I[S]= \bigcup_{u,v\in S} I[u,v]$. A subset $S\subseteq V$ is {\em geodesically convex} if $I[S] = S$. In other words, a subset $S$ is convex if, for any $u,v \in S$ and for any shortest $(u,v)$-path $P$, $V(P) \subseteq S$. Given a subset $S\subseteq V$, the {\em convex hull} $I_h[S]$ of $S$ is the smallest convex set that contains $S$. We say that $S$ is a {\em hull set} of $G$ if $I_h[S] = V$. The size of a minimum hull set of $G$ is the {\em hull number} of $G$, denoted by $hn(G)$. The {\sc Hull Number} problem is to decide whether $hn(G)\leq k$, for a given graph $G$ and an integer $k$. Dourado {\it et al.} showed that this problem is NP-complete in general graphs. In this paper, we answer an open question of Dourado et al.~\cite{Douradoetal09} by showing that the {\sc Hull Number} problem is NP-hard even when restricted to the class of bipartite graphs. Then, we design polynomial time algorithms to solve the {\sc Hull Number} problem in several graph classes. First, we deal with the class of complements of bipartite graphs. Then, we generalize some results in~\cite{ACGSS11} to the class of $(q,q-4)$-graphs and to the class of cacti. Finally, we prove tight upper bounds on the hull numbers. In particular, we show that the hull number of an $n$-node graph $G$ without simplicial vertices is at most $1+\lceil \frac{3(n-1)}{5}\rceil$ in general, at most $1+\lceil \frac{n-1}{2}\rceil$ if $G$ is regular or has no triangle, and at most $1+\lceil \frac{n-1}{3}\rceil$ if $G$ has girth at least $6$.Dans cet article nous étudions une notion de convexité dans les graphes. Nous nous concentrons sur la question de la compléxité du calcul de l'enveloppe minimum d'un graphe dans le cas de diverses classes de graphes. Étant donné un graphe $G = (V,E)$, l'intervalle $I[u,v]$ entre deux sommets $u,v \in V$ est l'ensemble des sommets qui appartiennent à un plus court chemin entre $u$ et $v$. Pour un ensemble $S\subseteq V$, on note $I[S]$ l'ensemble $\bigcup_{u,v\in S} I[u,v]$. Un ensemble $S\subseteq V$ de sommets est dit {\it convexe} si $I[S] = S$. L'{\it enveloppe convexe} $I_h[S]$ d'un sous-ensemble $S\subseteq V$ de $G$ est défini comme le plus petit ensemble convexe qui contient $S$. $S\subseteq V$ est une {\it enveloppe} de $G$ si $I_h[S] = V$. Le {\it nombre enveloppe} de $G$, noté $hn(G)$, est la cardinalité minimum d'une enveloppe de graphe $G$. Nous montrons que décider si $hn(G) \leq k$ est un problème NP-complet dans la classe des graphes bipartis et nous prouvons que $hn(G)$ peut être calculé en temps polynomial pour les cobipartis, $(q,q-4)$-graphes et cactus. Nous montrons aussi des bornes supérieures du nombre enveloppe des graphes en général, des graphes sans triangles et des graphes réguliers

On the hull number of some graph classes

International audienceIn this paper, we study the geodetic convexity of graphs focusing on the problem of the complexity to compute a minimum hull set of a graph in several graph classes. For any two vertices $u,v\in V$ of a connected graph $G=(V,E)$, the closed interval $I[u,v]$ of $u$ and $v$ is the the set of vertices that belong to some shortest $(u,v)$-path. For any $S \subseteq V$, let $I[S]= \bigcup_{u,v\in S} I[u,v]$. A subset $S\subseteq V$ is geodesically convex or convex if $I[S] = S$. In other words, a subset $S$ is convex if, for any $u,v \in S$ and for any shortest $(u,v)$-path $P$, $V(P) \subseteq S$. Given a subset $S\subseteq V$, the convex hull $I_h[S]$ of $S$ is the smallest convex set that contains $S$. We say that $S$ is a hull set of $G$ if $I_h[S] = V$. The size of a minimum hull set of $G$ is the hull number of $G$, denoted by $hn(G)$. The {\sc Hull Number} problem is to decide whether $hn(G)\leq k$, for a given graph $G$ and an integer $k$. Dourado {\it et al.} showed that this problem is NP-complete in general graphs. In this paper, we answer an open question of Dourado {\it et al.}~\cite{Douradoetal09} by showing that the {\sc Hull Number} problem is NP-hard even when restricted to the class of bipartite graphs. Then, we design polynomial time algorithms to solve the {\sc Hull Number} problem in several graph classes. First, we deal with the class of complements of bipartite graphs. Then, we generalize some results in~\cite{ACGSS11} to the class of $(q,q-4)$-graphs and to cacti. Finally, we prove tight upper bounds on the hull numbers. In particular, we show that the hull number of an $n$-node graph $G$ without simplicial vertices is at most $1+\lceil \frac{3(n-1)}{5}\rceil$ in general, at most $1+\lceil \frac{n-1}{2}\rceil$ if $G$ is regular or has no triangle, and at most $1+\lceil \frac{n-1}{3}\rceil$ if $G$ has girth at least $6$

On interval number in cycle convexity

International audienceRecently, Araujo et al. [Manuscript in preparation, 2017] introduced the notion of Cycle Convexity of graphs. In their seminal work, they studied the graph convexity parameter called hull number for this new graph convexity they proposed, and they presented some of its applications in Knot theory. Roughly, the tunnel number of a knot embedded in a plane is upper bounded by the hull number of a corresponding planar 4-regular graph in cycle convexity. In this paper, we go further in the study of this new graph convexity and we study the interval number of a graph in cycle convexity. This parameter is, alongside the hull number, one of the most studied parameters in the literature about graph convexities. Precisely, given a graph G, its interval number in cycle convexity, denoted by $in_{cc} (G)$, is the minimum cardinality of a set S ⊆ V (G) such that every vertex w ∈ V (G) \ S has two distinct neighbors u, v ∈ S such that u and v lie in same connected component of G[S], i.e. the subgraph of G induced by the vertices in S.In this work, first we provide bounds on $in_{cc} (G)$ and its relations to other graph convexity parameters, and explore its behavior on grids. Then, we present some hardness results by showing that deciding whether $in_{cc} (G)$ ≤ k is NP-complete, even if G is a split graph or a bounded-degree planar graph, and that the problem is W[2]-hard in bipartite graphs when k is the parameter. As a consequence, we obtainthat $in_{cc} (G)$ cannot be approximated up to a constant factor in the classes of split graphs and bipartite graphs (unless P = N P ).On the positive side, we present polynomial-time algorithms to compute $in_{cc} (G)$ for outerplanar graphs, cobipartite graphs and interval graphs. We also present fixed-parameter tractable (FPT) algorithms to compute it for (q, q − 4)-graphs when q is the parameter and for general graphs G when parameterized either by the treewidth or the neighborhood diversity of G.Some of our hardness results and positive results are not known to hold for related graph convexities and domination problems. We hope that the design of our new reductions and polynomial-time algorithms can be helpful in order to advance in the study of related graph problems

Caractérisation des instances difficiles de problèmes d'optimisation NP-difficiles

L'étude expérimentale d'algorithmes est un sujet crucial dans la conception de nouveaux algorithmes, puisque le contexte d'évaluation influence inévitablement la mesure de la qualité des algorithmes. Le sujet particulier qui nous intéresse dans l'étude expérimentale est la pertinence des instances choisies pour servir de base de test à l'expérimentation. Nous formalisons ce critère par la notion de "difficulté d'instance" qui dépend des performances pratiques de méthodes de résolution. Le coeur de la thèse porte sur un outil pour évaluer empiriquement la difficulté d'instance. L'approche proposée présente une méthode de benchmarking d'instances sur des jeux de test d'algorithmes. Nous illustrons cette méthode expérimentale pour évaluer des classes d'instances à travers plusieurs exemples d'applications sur le problème du voyageur de commerce. Nous présentons ensuite une approche pour générer des instances difficiles. Elle repose sur des opérations qui modifient les instances, mais qui permettent de retrouver facilement une solution optimale, d'une instance à l'autre. Nous étudions théoriquement et expérimentalement son impact sur les performances de méthodes de résolution.The empirical study of algorithms is a crucial topic in the design of new algorithms because the context of evaluation inevitably influences the measure of the quality of algorithms. In this topic, we particularly focus on the relevance of instances forming testbeds. We formalize this criterion with the notion of 'instance hardness' that depends on practical performance of some resolution methods. The aim of the thesis is to introduce a tool to evaluate instance hardness. The approach uses benchmarking of instances against a testbed of algorithms. We illustrate our experimental methodology to evaluate instance classes through several applications to the traveling salesman problem. We also suggest possibilities to generate hard instances. They rely on operations that modify instances but that allow to easily find the optimal solution of one instance from the other. We theoretically and empirically study their impact on the performance of some resolution methods.SAVOIE-SCD - Bib.électronique (730659901) / SudocGRENOBLE1/INP-Bib.électronique (384210012) / SudocGRENOBLE2/3-Bib.électronique (384219901) / SudocSudocFranceF