A note on vertices contained in the minimum dominating set of a graph with minimum degree three

Abstract

The minimum dominating set problem asks for a dominating set with minimum size. This paper shows how to determine some vertices contained in the minimum dominating set of a graph with minimum degree at least 3. Applying a particular scheme to the graph with minimum degree at least 3, the resulting graph is 2-connected and the length of each induced cycle is 0 mod 3. Label every three vertices for as many induced cycles as possible. Then there is a way of labeling in which the set of all labeled vertices is the minimum dominating set of the resulting graph, and is contained in the minimum dominating set of the original graph. The complexity of the minimum dominating set problem for cubic graphs was shown to be APX-complete in 2000 and this problem is partially resolved by our arguments.