On a Family of 4-Critical Graphs with Diameter Three

A graph G is k-total domination edge critical, abbreviated to k-critical if confusion is unlikely, if the total domination number gamma(t)(G) satisfies gamma(t)(G) = k and gamma(t)(G + e) < gamma(t)(G) for any edge e is an element of E(<(G)over bar>). Graphs that are 4-critical have diameter either 2, 3 or 4. In previous papers we characterized structurally the 4-critical graphs with diameter four, and found bounds on the order of 4-critical graphs with diameter two. In this paper we study a family H of 4-critical graphs with diameter three, in which every vertex is a diametrical vertex, and every diametrical pair dominates the graph. We also generalize the self-complementary graphs, and show that these graphs provide a special case of the family H.