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Reports (Research Report) Year : 2011

On the hull number of some graph classes

Abstract

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.
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Dates and versions

inria-00576581 , version 1 (14-03-2011)
inria-00576581 , version 2 (14-09-2011)

Identifiers

  • HAL Id : inria-00576581 , version 2

Cite

Julio Araujo, Victor Campos, Frédéric Giroire, Nicolas Nisse, Leonardo Sampaio, et al.. On the hull number of some graph classes. [Research Report] RR-7567, INRIA. 2011, pp.19. ⟨inria-00576581v2⟩
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