The bondage and reinforcement numbers of gammaf for some graphs

Abstract For any graph G , a real-valued function g : V ( G ) → [0, 1] is called a dominating function if for every v ϵ V ( G ), Σ wϵN [ v ] g ( w ) ⩾ 1 The fractional domination number is defined to be γ f ( G ) =min{ Σ vϵV ( G ) g ( v ): g is a dominating function of G }. In this paper, we initiate the study of bondage and reinforcement associated with fractional domination. The bondage number of γ f , b f ( G ), is defined to be the minimum cardinality of a set of edges whose removal from G results in a graph G 1 satisfying γ f ( G 1 ) > γ f ( G ). The reinforcement number of γ f , r f ( G ), is defined to be the minimum cardinality of a set of edges which when added to G results in a graph G ″ satisfying γ f ( G ″) γ f ( G ). We will give exact values of b f ( G ) and r f ( G ) for some classes of graphs.