On the geodetic and the hull numbers in strong product graphs

Cáceres, José; Hernando, Carmen; Mora, Mercè; Pelayo, Ignacio M.; Puertas, María Luz

doi:10.1016/j.camwa.2010.10.001

Cited by 38 publications

(17 citation statements)

References 19 publications

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“…Bounds on the hull number of trianglefree graphs are shown in [13]. 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.…”

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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“…Two distinct vertices (v i , u j ) and (v k , u l ) of G H are adjacent if and only if (v i = v k and u j ∼ u l ), or (v i ∼ v k and u j = u l ), or (v i ∼ v k and u j ∼ u l ). The following results on geodetic sets in strong product graphs were obtained in [4]. Lemma 2.20.…”

Section: Product Graphsmentioning

confidence: 99%

“…[4] If P is a shortest (a, b) − (c, d) path in a strong product graph G H such that d G H ((a, b), (c, d)) = d G (a, b) = p, then the projection of P over the graph G is a shortest a − c path of length p. Lemma 2.21. [4] For any two graphs G and H, min{ (G), (H)} ≤ (G H) ≤ (G) (H). Lemma 2.22.…”

Section: Product Graphsmentioning

confidence: 99%

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Convex dominating-geodetic partitions in graphs

Yero

Lemańska

2016

Filomat

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The distance d(u, v) between two vertices u and v in a connected graph G is the length of aThe convex domination number γ con (G) of a graph G equals the minimum cardinality of a convex dominating set in G. A set of vertices S of a graph G is a geodetic set of G if every vertex v S lies on a x − y geodesic between two vertices x, y of S. The minimum cardinality of a geodetic set of G is the geodetic number of G and it is denoted by (G). Let D, S be a convex dominating set and a geodetic set in G, respectively. The two sets D and S form a convex dominating-geodetic partition of G if |D| + |S| = |V(G)|. Moreover, a convex dominating-geodetic partition of G is called optimal if D is a γ con (G)-set and S is a (G)-set. In the present article we study the (optimal) convex dominating-geodetic partitions of graphs.

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“…This means that the calculation of the geodetic and the hull numbers of strong product graphs of the form G C n requires a different approach to the previous one. In Table 3.3, both the hull and the geodetic number of some basic families of the form G C n is displayed [27].…”

Section: G H Is An Extreme Geodesic Graph If and Only If Both G And Hmentioning

confidence: 99%

Geodesic Convexity in Graphs

Pelayo¹

2013

SpringerBriefs in Mathematics

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SpringerBriefs in Mathematics showcases expositions in all areas of mathematics and applied mathematics. Manuscripts presenting new results or a single new result in a classical field, new field, or an emerging topic, applications, or bridges between new results and already published works, are encouraged. The series is intended for mathematicians and applied School of Agricultural Engineering of Barcelona

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On the geodetic and the hull numbers in strong product graphs

Cited by 38 publications

References 19 publications

On the hull number of some graph classes

On the hull number of some graph classes

Convex dominating-geodetic partitions in graphs

Geodesic Convexity in Graphs

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