2021/04/14 by Zhuang, Wei
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2104.07026
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, γd2(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 γd2(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.