On 3-component domination numbers in graphs

Abstract

Let $s$ be a positive integer and let $G=(V\left(G\right),E\left(G\right))$ be a graph. A vertex set $D$ is an $s$-component dominating set of $G$ if every vertex outside $D$ has a neighbor in $D$ and every component of the subgraph induced by $D$ in $G$ contains at least $s$ vertices. The minimum cardinality of an $s$-component dominating set of $G$ is the $s$-component domination number ${\gamma }_s\left(G\right)$ of $G$. Determining the exact values or bounds of domination parameters on graphs is an important, basic, and challenging problem in the graph domination field. The tree $T$ and the generalized Petersen graph $P(n,k)$ with $k\ge 1$ are the significant graph classes in graph theory. In this paper, we first give an upper bound of the 3-component domination number of a tree $T$. Then, we study the $s$-component domination numbers on $P(n,k)$ and get the exact values of 3-component domination numbers on $P(n,1)$ and $P(n,2)$.