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

Abstract

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