On α-Domination in Graphs

Let G = ( V , E ) be an isolate-free graph. For some α with 0 < α ≤ 1 , a subset S of V is said to be an α -dominating set if for all v ∈ V \ S , | N ( v ) ∩ S | ≥ α | N ( v ) | . The size of a smallest such S is called the α -domination number and is denoted by γ α ( G ) . A set S ⊆ V is said to be an α -rate dominating set of G if for any vertex v ∈ V , | N [ v ] ∩ X | ≥ α | N ( v ) | . The minimum cardinality of an α -rate dominating set of G is called the α -rate domination number γ × α ( G ) . The set of distinct values of γ α ( G ) as α runs over (0, 1] is called the α -domination spectrum of a graph G , i.e., Sp α ( G ) = { γ α ( G ) : α ∈ ( 0 , 1 ] } . In this paper, we study some properties of Sp α ( G ) and show that γ α ( G ) changes its value only at rational points as α runs over (0, 1]. Using this result, we characterize some values of α such that γ α ( G ) ≤ n α , where n is the number of vertices in G , holds. Finally, we present some improved probabilistic upper bounds of α -domination number and α -rate domination number of a graph G .