Uniformly pair-bonded trees

Let G = (V(G), E(G)) be a graph with δ(G) ≥ 1. A set D ⊆ V(G) is a paired-dominating set if D is a dominating set and the induced subgraph G[D] contains a perfect matching. The paired domination number of G, denoted by ϒᵨ(G), is the minimum cardinality of a paired-dominating set of G. The paired bondage number, denoted by bᵨ(G), is the minimum cardinality among all sets of edges Eʹ ⊆ E such that δ(G-Eʹ) ≥ 1 and ϒᵨ(G-Eʹ) ˃ ϒᵨ(G). For any bᵨ(G) edges Eʹ ⊆ E with δ(G-Eʹ) ≥ 1, if ϒᵨ(G-Eʹ) ˃ ϒᵨ(G), then G is called uniformly pair-bonded graph. In this paper, we prove that there exists uniformly pair-ponded tree T with bᵨ(T) = k for any positive integer k. Furthermore, we give a constructive characterization of uniformly pair-bonded trees. Copyright © 2010 Dept. of Combinatorics and Optimization, University of Waterloo