Domination dot-critical graphs with no critical vertices

A graph GG is dot-critical if contracting any edge decreases the domination number. It is totally dot-critical if identifying any two vertices decreases the domination number. Burton and Sumner [Discrete Math. 306 (2006) 11–18] posed the problem: Is it true that for k⩾4k⩾4, there exists a totally kk-dot-critical graph with no critical vertices? In this paper, we show that this problem has a positive answer.