Paired Domination in Trees

Abstract

A set S of vertices in a graph G is a paired dominating set if every vertex of G is adjacent to a vertex in S and the subgraph induced by S contains a perfect matching (not necessarily as an induced subgraph). The paired domination number, gamma _{mathrm{pr}}(G), of G is the minimum cardinality of a paired dominating set of G. In this paper, we show that if T is a tree of order at least 2, then gamma _{mathrm{pr}}(T) le 2alpha (T) - varphi (T) where alpha (T) is the independence number and varphi (T) is the P_3-packing number. We present a tight upper bound on the paired domination number of a tree T in terms of its maximum degree varDelta. For varDelta ge 1, we show that if T is a tree of order n with maximum degree varDelta, then gamma _{mathrm{pr}}(T) le left( frac{5varDelta -4}{8varDelta -4} right) n + frac{1}{2}n_1(T) + frac{1}{4}n_2(T) - left( frac{varDelta -2}{4varDelta -2} right), where n_1(T) and n_2(T) denote the number of vertices of degree 1 and 2, respectively, in T. Further, we show that this bound is tight for all varDelta ge 3. As a consequence of this result, if T is a tree of order n ge 2, then gamma _{mathrm{pr}}(T) le frac{5}{8} n + frac{1}{2}n_1(T) + frac{1}{4}n_2(T), and this bound is asymptotically best possible.