On the hull number of some graph classes

Abstract

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 ∈ 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 shortestIn other words, a subset S is convex if, for any u, v ∈ S and for any shortest (u, v)-path P, V (P) ⊆ S. Given a subset S ⊆ 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 HULL NUMBER problem is to decide whether hn(G) ≤ k, for a given graph G and an integer k. Dourado et al. showed that this problem is NP-complete in general graphs.In this paper, we answer an open question of Dourado et al. [12] by showing that the HULL NUMBER problem is NP-hard even when restricted to the class of bipartite graphs. Then, we design polynomial time algorithms to solve the HULL NUMBER problem in several graph classes. First, we deal with the class of complements of bipartite graphs. Then, we generalize some results in [1] 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 + ⌈ Le nombre enveloppe de quelques classes de graphesRésumé : Dans cet article nousétudions 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. Etant donné un graphe G = (V, E), l'intervalle I [u, v] entre deux sommets u, v ∈ V est l'ensemble des sommets qui appartiennentà un plus court chemin entre u et v. Pour un ensemble S ⊆ V , on note I [S] l'ensemble u,v∈S I[u, v].Nous montrons que décider si hn(G) ≤ k est un problème NP-complet dans la classe des graphes bipartis et nous prouvons que hn(G) peutêtre 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.

Inapproximability Results for Graph Convexity Parameters

“…Regarding the geodetic convexity, it was proved that the same parameters are also NP-hard [1,14,15,18].…”

“…On the other way, given a P 3 -interval set C of G with at least 5 vertices, it is possible to prove that we can easily obtain a good P 3 -interval set C with |C| ≤ |C |, which is a set C that contains {x 1 , x 2 , x 3 , x 4 } and C \ {x 1 , x 2 , x 3 , x 4 } ⊆ S. Let g(S, C) be the sets of S associated to the vertices in C \ {x 1 …”

On the geodetic hull number of P-free graphs

“…The hull number is NP-hard for bipartite graphs [2] and even for partial cubes [1], but can be computed in polynomial time for cographs [9], (q, q − 4)-graphs [2], {C 3 , P 5 }-free graphs [3], distance-hereditary graphs [21], and chordal graphs [21].…”

“…Bounds on the hull number are given in [2,10,17]. The interval number is NP-hard for cobipartite graphs [15] and for chordal graphs as well as for chordal bipartite graphs [11,20], but can be computed in polynomial time for split graphs [11], proper u u u u e e e ¡ ¡ ¡ u u u…”

The Geodetic Hull Number is Hard for Chordal Graphs

“…One of the most well studied convexity notions for graphs is the shortest path convexity or geodetic convexity, where a set X of vertices of a graph G is considered convex if no vertex outside of S lies on a shortest path between two vertices inside of S. Defining the convex hull of a set S of vertices as the smallest convex set containing S, a natural parameter of G is its hull number h(G) [7], which is the minimum order of a set of vertices whose convex hull is the entire vertex set of G. The hull number is NP-hard for bipartite graphs [2], partial cubes [1], and P 9 -free graphs [5], but it can be computed in polynomial time for cographs [4], (q, q − 4)-graphs [2], {paw, P 5 }-free graphs [3,5], and distance-hereditary graphs [9]. Bounds on the hull number are given in [2,6,7].…”