Perfect and quasiperfect domination in trees

Abstract

A k-quasiperfect dominating set of a connected graph G is a vertex subset S such that 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 γ1k(G). These graph parameters were first introduced by Chellali et al. (2013) as a generalization of both the perfect domination number 11(G) and the domination number γ(G). The study of the so-called quasiperfect domination chain γ11(G) ≥ γ12(G) ≥ ... ≥γ1Δ(G) = γ(G) en- ables us to analyze how far minimum dominating sets are from being perfect. We provide, for any tree T and any positive integer k, a tight upper bound of γ1k(T). We also prove that there are trees satisfying all possible equalities and inequalities in this chain. Finally a linear algorithm for computing γ1k(T) in any tree T is presented.