Domination on hyperbolic graphs

Abstract

If k≥1 and G=(V,E) is a finite connected graph, S⊆V is said a distancek-dominating set if every vertex v∈V is within distance k from some vertex of S. The distancek-domination numberγwk(G) is the minimum cardinality among all distance k-dominating sets of G. A set S⊆V is a total dominating set if every vertex v∈V satisfies δS(v)≥1 and the total domination number, denoted by γt(G), is the minimum cardinality among all total dominating sets of G. The study of hyperbolic graphs is an interesting topic since the hyperbolicity of any geodesic metric space is equivalent to the hyperbolicity of a graph related to it. In this paper we obtain relationships between the hyperbolicity constant δ(G) and some domination parameters of a graph G. The results in this work are inequalities, such as γwk(G)≥2δ(G)∕(2k+1) and δ(G)≤γt(G)∕2+3.