Total Domination Edge Critical Graphs with Total Domination Number Three and Many Dominating Pairs

A graph is diameter-2-critical if its diameter is two and the deletion of any edge increases the diameter. Murty and Simon conjectured that the number of edges in a diameter-2-critical graph of order is at most and that the extremal graphs are the complete bipartite graphs . A graph is -edge-critical, abbreviated , if its total domination number is 3 and the addition of any edge decreases the total domination number. It is known that proving the Murty-Simon Conjecture is equivalent to proving that the number of edges in a graph of order is greater than . We study a family of graphs of diameter 2 for which every pair of nonadjacent vertices dominates the graph. We show that the graphs in are precisely the bull-free graphs and that the number of edges in such graphs is at least , proving the conjecture for this family. We characterize the extremal graphs, and conjecture that this improved bound is in fact a lower bound for all graphs of diameter 2. Finally we slightly relax the requirement in the definition of -instead of requiring that all pairs of nonadjacent vertices dominate to requiring that only most of these pairs dominate-and prove the Murty-Simon equivalent conjecture for these graphs.