On \(\alpha \)-Domination in Graphs

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