Codiameters of 3-domination critical graphs with toughness more than one

A graph G is 3-domination-critical (3-critical, for short), if its domination number γ is 3 and the addition of any edge decreases γ by 1. In this paper, we show that every 3-critical graph with independence number 4 and minimum degree 3 is Hamilton-connected. Combining the result with those in [2], [4] and [5], we solve the following conjecture: a connected 3critical graph G is Hamilton-connected if and only if τ(G) > 1, where τ(G) is the toughness of G.