3-component domination numbers in graphs.

Let k be a positive integer and let G = (V (G) , E (G)) be a graph. A vertex set D is a k -component dominating set of G if every vertex outside D in G has a neighbor in D and every component of the subgraph G [ D ] of G induced by D contains at least k vertices. The minimum cardinality of a k -component dominating set of G is the k -component domination number γ k (G) of G. It was conjectured that if G is a connected graph of order n ≥ k + 1 , and minimum degree at least 2, then γ k (G) ≤ 2 k n 2 k + 3 except for a finite set of graphs. In this paper, we focus on the parameter γ 3 (G) of G. We first determine the exact values of 3-component domination numbers of paths and cycles. We then proceed to show that if G is a connected graph of order n with minimum degree at least 2 and maximum degree at most 3, then γ 3 (G) ≤ 2 n 3 , unless G is one of seven special graphs. This result provides positive support for the conjecture and also generalizes a result by Alvarado et al. (2016) [1]. [ABSTRACT FROM AUTHOR]