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 [Formula presented] except for a finite set of graphs. In this paper, we focus on the parameter γ₃(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 [Formula presented], 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].