Uniformly most reliable three-terminal graph of dense graphs

Suppose that every edge of a graph $G$ survives independently with a fixed probability between $0$ and $1$. The three-terminal reliability is the connection probability of the fixed three target vertices $r,s$ and $t$ in a three-terminal graph. This research focuses on the uniformly most reliable three-terminal graph of dense graphs with $n$ vertices and $m$ edges in some ranges. First, we give the locally most reliable three-terminal graphs of $n$ and $m$ in a certain range for $p$ close to $0$ and for $p$ close to $1$. And then, we prove that there is no uniformly most reliable three-terminal graph with certain ranges of $n$ and $m$. Finally, some uniformly most reliable graphs are given for $\binom{n}{2}-2$ ($4\leq n\leq 6$) and $\binom{n}{2}-1$ ($n\geq5$). This study of uniformly or locally most reliable three-terminal graph provides helpful guidance for constructing highly reliable network structures involving three key vertices as target vertices