Disjunctive domination in graphs with minimum degree at least two

Abstract

A set D of vertices in G is a disjunctive dominating set in G if every vertex not in D is adjacent to a vertex of D or has at least two vertices in D at distance 2 from it in G. The disjunctive domination number, γ2d(G), of G is the minimum cardinality of a disjunctive dominating set in G. In this paper, we show that if G be a graph of order at least 3, δ(G)≥2 and with no component isomorphic to any of eight forbidden graphs, then γ2d(G)≤|G|3. Moreover, we provide an infinite family of graphs attaining this bound. In addition, we also study the case that G is a claw-free graph with minimum degree at least two.