On $\alpha$-Domination in Graphs

Abstract

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.