On Perfect and Quasiperfect Dominations in Graphs∗

A subset 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 γ 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 ≥ γ11(G) ≥ γ12(G) ≥ . . . ≥ γ1∆(G) = γ(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, γ12(G) = γ(G). Among them, one can find cographs, claw-free graphs and graphs with extremal values of ∆(G).