Complexity results for $k$-domination and $α$-domination problems and their variants

Abstract

Let $G=(V, E)$ be a simple and undirected graph. For some integer $k\geq 1$, a set $D\subseteq V$ is said to be a k-dominating set in $G$ if every vertex $v$ of $G$ outside $D$ has at least $k$ neighbors in $D$. Furthermore, for some real number $α$ with $0<α\leq1$, a set $D\subseteq V$ is called an $α$-dominating set in $G$ if every vertex $v$ of $G$ outside $D$ has at least $α\times d_v$ neighbors in $D$, where $d_v$ is the degree of $v$ in $G$. The cardinality of a minimum $k$-dominating set and a minimum $α$-dominating set in $G$ is said to be the $k$-domination number and the $α$-domination number of $G$, respectively. In this paper, we present some approximability and inapproximability results on the problem of finding $k$-domination number and $α$-domination number of some classes of graphs. Moreover, we introduce a generalization of $α$-dominating set which we call an $f$-dominating set. Given a function $f:\mathbb{N}\rightarrow \mathbb{R}$, where $\mathbb{N}=\{1, 2, 3, \ldots\}$, a set $D\subseteq V$ is said to be an $f$-dominating set in $G$ if every vertex $v$ of $G$ outside $D$ has at least $f(d_v)$ neighbors in $D$. We prove NP-hardness of the problem of finding a minimum $f$-dominating set in $G$, for a large family of functions $f$.