The connectivity of domination dot-critical graphs with no critical vertices

An edge of a graph is called dot-critical if its contraction decreases the domination number. A graph is said to be dot-critical if all of its edges are dot-critical. A vertex of a graph is called critical if its deletion decreases the domination number. In A note on the domination dot-critical graphs, Discrete Appl. Math. 157 (2009) 3743–3745, Chen and Shiu constructed for each even integer k ≥ 4 infinitely many k-dot-critical graphs G with no critical vertices and κ(G) = 1. In this paper, we refine their result and construct for integers k ≥ 4 and l ≥ 1 infinitely many k-dot-critical graphs G with no critical vertices, κ(G) = 1 and λ(G) = l. Furthermore, we prove that every 3-dotcritical graph with no critical vertices is 3-connected, and it is best possible.