γ-Paired dominating graphs of lollipop, umbrella and coconut graphs

Abstract

A paired dominating set of a graph G is a dominating set whose induced subgraph has a perfect matching. The paired domination number γ p r ( G ) of G is the minimum cardinality of a paired dominating set. A paired dominating set D is a γ p r ( G ) -set if | D |= γ p r ( G ) . The γ -paired dominating graph P D γ ( G ) of G is the graph whose vertex set is the set of all γ p r ( G ) -sets, and two γ p r ( G ) -sets D 1 and D 2 are adjacent in P D γ ( G ) if D 2 = ( D 1 \ { u }) ∪ { v } for some u ∈ D 1 and v ∉ D 1 . This paper determines the paired domination numbers of lollipop graphs, umbrella graphs, and coconut graphs. We also consider the γ -paired dominating graphs of those three graphs.