Total domination versus paired domination

Abstract

A dominating set of a graph G is a vertex subset that any vertex of G either belongs to or is adjacent to. A total dominating set is a dominating set whose induced subgraph does not contain isolated vertices. The minimal size of a total dominating set, the total domination number, is denoted by t. The maximal size of an inclusionwise minimal total dominating set, the upper total domination number, is denoted by t. A paired dominating set is a dominating set whose induced subgraph has a perfect matching. The minimal size of a paired dominating set, the paired domination number, is denoted by p. The maximal size of an inclusionwise minimal paired dominating set, the upper paired domination number, is denoted by p. In this paper we prove several results on the ratio of these four parameters: For each r ≥ 2 we prove the sharp bound p/t ≤ 2−2/r for K1,r-free graphs. As a consequence, we obtain the sharp bound p/t ≤ 2−2/(�+1),