Paired bondage in trees

Let G=(V,E)G=(V,E) be a graph with δ(G)≥1δ(G)≥1. A set D⊆VD⊆V is a paired dominating set   if DD is dominating, and the induced subgraph 〈D〉〈D〉 contains a perfect matching. The paired domination number   of GG, denoted by γp(G)γp(G), is the minimum cardinality of a paired dominating set of GG. The paired bondage number  , denoted by bp(G)bp(G), is the minimum cardinality among all sets of edges E′⊆EE′⊆E such that δ(G−E′)≥1δ(G−E′)≥1 and γp(G−E′)>γp(G)γp(G−E′)>γp(G). We say that GG is a γpγp-strongly stable graph   if, for all E′⊆EE′⊆E, either γp(G−E′)=γp(G)γp(G−E′)=γp(G) or δ(G−E′)=0δ(G−E′)=0. We discuss the basic properties of paired bondage and give a constructive characterization of γpγp-strongly stable trees.