Perfect and quasiperfect domination in trees

A k-quasiperfect dominating set 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 gamma(1k)(G). Those sets were first introduced by CHELLALI et al. (2013) as a generalization of both the perfect domination concept. The quasiperfect domination chain gamma(11)(G) >= gamma(12)(G) >= ... >= gamma(1 Delta)(G) = gamma(G), indicates what it is lost in size when you move towards a more perfect domination. We provide an upper bound for gamma(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 gamma(1k)(T) in any tree is presented.