Paired bondage in trees

Abstract

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