Paired domination stability in graphs

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, γpr(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 γpr−-stability of G, denoted stγpr−(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 γpr(G) ≥ 4, then stγpr−(G) ≤ 2Δ(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.