Least domination in a graph Original Research Article

The least domination number γL of a graph G is the minimum cardinality of a dominating set of G whose domination number is minimum. The least point covering number αL of G is the minimum cardinality of a total point cover (point cover including every isolated vertex of G) whose total point covering number is minimum. We prove a conjecture of Sampathkumar saying that γL ⩽ 35n in every connected graph of order n ⩾ 2. We disprove another one saying that γL ⩽ αL in every graph but instead of it, we establish the best possible inequality γL ⩽ 32 αL. Finally, in relation with the minimum cardinality γt of a dominating set without isolated vertices (total dominating set), we prove that the ratio γL/γt can be in general arbitrarily large, but remains bounded by 95 if we restrict ourselves to the class of trees.