A graph G is said to be bicritical if the removal of any pair of vertices decreases the domination number of G. For a bicritical graph G with the domination number t, we say that G is t-bicritical. Let λ(G) denote the edge-connectivity of G. In [2], Brigham et al. (2005) posed the following question: If G is a connected bicritical graph, is it true that λ(G)≥3?In this paper, we give a negative answer toward this question; namely, we give a construction of infinitely many connected t-bicritical graphs with edge-connectivity 2 for every integer t≥5. Furthermore, we give some sufficient conditions for a connected 5-bicritical graph to have λ(G)≥3.