On the hull number of some graph classes

Julio Araujo 1, 2 Victor Campos 1, 2 Frédéric Giroire 2 Nicolas Nisse 2 Leonardo Sampaio 2 Ronan Soares 1, 2
2 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
Abstract : In 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 $6$.
Complete list of metadatas

Cited literature [18 references]  Display  Hide  Download

https://hal.inria.fr/hal-00770650
Contributor : Julio Araujo <>
Submitted on : Monday, January 7, 2013 - 12:07:34 PM
Last modification on : Friday, April 12, 2019 - 10:18:09 AM
Long-term archiving on : Monday, April 8, 2013 - 11:20:39 AM

File

Hull-TCS-Corrected.pdf
Files produced by the author(s)

Identifiers

Collections

Citation

Julio Araujo, Victor Campos, Frédéric Giroire, Nicolas Nisse, Leonardo Sampaio, et al.. On the hull number of some graph classes. Theoretical Computer Science, Elsevier, 2013, 475, pp.1-12. ⟨10.1016/j.tcs.2012.12.035⟩. ⟨hal-00770650⟩

Share

Metrics

Record views

484

Files downloads

246