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On the hull number of some graph classes

Julio Araújo 1 Victor Campos 2 Frédéric Giroire 1 Leonardo Sampaio 1 Ronan Pardo Soares 1
1 MASCOTTE - Algorithms, simulation, combinatorics and optimization for telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
Abstract : Given a graph $G = (V,E)$, the {\em closed interval} of a pair of vertices $u,v \in V$, denoted by $I[u,v]$, is the set of vertices that belongs to some shortest $(u,v)$-path. For a given $S\subseteq V$, let $I[S] = \bigcup_{u,v\in S} I[u,v]$. We say that $S\subseteq V$ is a {\em convex set} if $I[S] = S$. The {\em convex hull} $I_h[S]$ of a subset $S\subseteq V$ is the smallest convex set that contains $S$. We say that $S$ is a {\em hull set} if $I_h[S] = V$. The cardinality of a minimum hull set of $G$ is the {\em hull number} of $G$, denoted by $hn(G)$. We show that deciding if $hn(G)\leq k$ is an NP-complete problem, even if $G$ is bipartite. We also prove that $hn(G)$ can be computed in polynomial time for cactus and $P_4$-sparse graphs.
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Submitted on : Monday, October 24, 2011 - 3:34:54 PM
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Julio Araújo, Victor Campos, Frédéric Giroire, Leonardo Sampaio, Ronan Pardo Soares. On the hull number of some graph classes. EuroComb'11 - European Conference on Combinatorics, Graph Theory and Applications, Rényi Institute, Aug 2011, Budapest, Hungary. pp.49-55, ⟨10.1016/j.endm.2011.09.009⟩. ⟨inria-00635032⟩

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