Perfect and Quasiperfect Domination in Trees

A k−quasiperfect dominating set (k ≥ 1) of a 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 γ 1k (G). Those sets were first introduced by Chellali et al. (2013) as a generalization of the perfect domination concept. The quasiperfect domination chain γ11(G) ≥ γ12(G) ≥ · · · ≥ γ1∆(G) = γ(G), indicates what it is lost in size when you move towards a more perfect domination. We provide an upper bound for γ 1k (T ) in any tree T and trees achieving this bound are characterized. We prove that there exist trees satisfying all the possible equalities and inequalities in this chain and a linear algorithm for computing γ 1k (T ) in any tree is presented.