On Perfect and Quasiperfect Dominations in Graphs

A subset S subset of 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 gamma(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 >= gamma(11) (G) >= gamma(12) (G) >= ... >= gamma(1 Delta)(G) = gamma (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, gamma(12)(G) = gamma(G). Among them, one can find cographs, claw-free graphs and graphs with extremal values of Delta(G).