On 3-component domination numbers in graphs

Let s be a positive integer and let G=(V(G),E(G)) 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 γ_s(G) 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≥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).