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

Abstract

A graph $$G$$G 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 $$G$$G of order $$n$$n is at most $$\lfloor n^2/4 \rfloor $$?n2/4? and that the extremal graphs are the complete bipartite graphs $$K_{{\lfloor n/2 \rfloor },{\lceil n/2 \rceil }}$$K?n/2?,?n/2?. A graph is $$3_t$$3t-edge-critical, abbreviated $$3_tEC$$3tEC, 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 $$3_tEC$$3tEC graph of order $$n$$n is greater than $$\lceil n(n-2)/4 \rceil $$?n(n-2)/4?. We study a family $$\mathcal{F}$$F of $$3_tEC$$3tEC graphs of diameter 2 for which every pair of nonadjacent vertices dominates the graph. We show that the graphs in $$\mathcal{F}$$F are precisely the bull-free $$3_tEC$$3tEC graphs and that the number of edges in such graphs is at least $$\lfloor (n^2 - 4)/4 \rfloor $$?(n2-4)/4?, 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 $$3_tEC$$3tEC graphs of diameter 2. Finally we slightly relax the requirement in the definition of $$\mathcal{F}$$F--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 $$3_tEC$$3tEC graphs.