On the Steiner, geodetic and hull numbers of graphs

Hernando, Carmen; Jiang, Tao; Mora, Mercè; Pelayo, Ignacio M.; Seara, Carlos

doi:10.1016/j.disc.2004.08.039

Cited by 72 publications

(48 citation statements)

References 11 publications

(13 reference statements)

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“…The hull number of the cartesian and the strong product of two connected graphs is studied in [5,11]. In [18], the authors have studied the relationship between the Steiner number and the hull number of a given graph. An oriented version of the HULL NUMBER problem is studied in [8,17].…”

Section: For Any S ⊆ V Let I[s] = Uv∈s I[u V] a Subset S ⊆ V Is mentioning

confidence: 99%

On the hull number of some graph classes

Araújo

Campos

Giroire

et al. 2011

Electronic Notes in Discrete Mathematics

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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.

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Section: For Any S ⊆ V Let I[s] = Uv∈s I[u V] a Subset S ⊆ V Is mentioning

confidence: 99%

On the hull number of some graph classes

Araújo

Campos

Giroire

et al. 2011

Electronic Notes in Discrete Mathematics

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“…As for the second generalization, consisting of using the geodetic closure operator, a number of results have recently been obtained [2,3,7,8]. For example, it has been proved that in the class of distance-hereditary graphs, every convex set is the geodetic closure of its contour vertices [3,8].…”

Section: Introductionmentioning

confidence: 98%

Geodeticity of the contour of chordal graphs

Cáceres

Hernando

Mora

et al. 2008

Discrete Applied Mathematics

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A vertex v is a boundary vertex of a connected graph G if there exists a vertex u such that no neighbor of v is further away from u than v. Moreover, if no vertex in the whole graph V (G) is further away from u than v, then v is called an eccentric vertex of G. A vertex v belongs to the contour of G if no neighbor of v has an eccentricity greater than the eccentricity of v. Furthermore, if no vertex in the whole graph V (G) has an eccentricity greater than the eccentricity of v, then v is called a peripheral vertex of G. This paper is devoted to study these kinds of vertices for the family of chordal graphs. Our main contributions are, firstly, obtaining a realization theorem involving the cardinalities of the periphery, the contour, the eccentric subgraph and the boundary, and secondly, proving both that the contour of every chordal graph is geodetic and that this statement is not true for every perfect graph.

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“…The geodetic number was first introduced in 1990 (see [4]) and is further studied in [1][2][3][5][6][7][8] and [9]. Several classes of graphs and their geodetic number are presented in [8].…”

Section: Introductionmentioning

confidence: 99%

On pitfalls in computing the geodetic number of a graph

Hansen

Omme

2006

Optimization Letters

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Given a simple connected graph G = (V, E) the geodetic closure I[S] ⊂ V of a subset S of V is the union of all sets of nodes lying on some geodesic (or shortest path) joining a pair of nodes v k , v l ∈ S. The geodetic number, denoted by g(G), is the smallest cardinality of a node set S * such that I[S * ] = V. In "The geodetic number of a graph", [Harary et al. in Math. Comput. Model. 17:89-95, 1993] propose an incorrect algorithm to find the geodetic number of a graph G. We provide counterexamples and show why the proposed approach must fail. We then develop a 0-1 integer programming model to find the geodetic number. Computational results are given.

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On the Steiner, geodetic and hull numbers of graphs

Cited by 72 publications

References 11 publications

On the hull number of some graph classes

On the hull number of some graph classes

Geodeticity of the contour of chordal graphs

On pitfalls in computing the geodetic number of a graph

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