Upper bounds on the diameter of domination dot-critical graphs with given connectivity

The domination number @c(G) of a graph G is the minimum cardinality of any dominating set of G. 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. In general, every connected graph G has diameter at most 3@c(G)-1. It is known that the diameter of a connected dot-critical graph G is at most 3@c(G)-3. In this paper, we show that, if a dot-critical graph G has diameter exactly 3@c(G)-3, then G is a path. Furthermore, we focus on dot-critical graphs with high connectivity. We prove that, for l>=2, the diameter of an l-connected dot-critical graph G is at most 2@c(G)-2, and show that the bound 2@c(G)-2 is best possible.