On perfect and quasiperfect dominations in graphs

Abstract

Asubset S ? V in a graph G=(V,E) is a k-quasiperfect dominating set (for k ? 1) if every vertex not in S is adjacent to at least one and at most k vertices in S. The cardinality of a minimum k-quasiperfect dominating set in G is denoted by 1k(G). Those sets were first introduced by Chellali et al. (2013) as a generalization of the perfect domination concept and allow us to construct a decreasing chain of quasiperfect dominating numbers n ? ?11(G) ? ?12(G) ?...? ?1?(G) = ?(G) in order to indicate how far is G from being perfectly dominated. In this paper we study properties, existence and realization of graphs for which the chain is short, that is, ?12(G) = ?(G). Among them, one can find cographs, claw-free graphs and graphs with extremal values of ?(G).