Paired domination stability in graphs

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, γ pr ( G ) , of G is the minimum cardinality of a paired dominating set of G . A set of vertices whose removal from G produces a graph without isolated vertices is called a non-isolating set. The minimum cardinality of a non-isolating set of vertices whose removal decreases the paired domination number is the γ pr − -stability of G , denoted st γ pr − ( G ) . The paired domination stability of G is the minimum cardinality of a non-isolating set of vertices in G whose removal changes the paired domination number. We establish properties of paired domination stability in graphs. We prove that if G is a connected graph with γ pr ( G ) ≥ 4 , then st γ pr − ( G ) ≤ 2 Δ ( G ) where Δ ( G ) is the maximum degree in G , and we characterize the infinite family of trees that achieve equality in this upper bound.