On 3-component domination numbers in graphs

Lets be a positive integer and let G = (V(G), E ( G )) be a graph. A vertex set D is an scomponent 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 scomponent domination number gamma 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). (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.