Quasiperfect Domination in Trees

A k–quasiperfect dominating set (  k≥1k≥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 of G   is denoted by γ1k(G)γ1k(G). Those sets were first introduced by Chellali et al. (2013) as a generalization of the perfect domination concept (which coincides with the case k=1k=1) and allow us to construct a decreasing chain of quasiperfect dominating parametersequation(1)γ11(G)≥γ12(G)≥…≥γ1,Δ(G)=γ(G),γ11(G)≥γ12(G)≥…≥γ1,Δ(G)=γ(G), in order to indicate how far is G from being perfectly dominated. In this work, we study general properties, tight bounds, existence and realization results involving the parameters of the so-called QP-chain ( 1), for trees.