Existence of uniformly most reliable two-terminal graphs with distance constraints

Source abstract

A two-terminal graph is a simple graph equipped with two distinguished vertices, called terminals. Let $T_{n,m}$ be the class consisting of all non-isomorphic two-terminal graphs on $n$ vertices and $m$ edges. Let $G$ be any two-terminal graph in $T_{n,m}$, and let $d$ be any positive integer. For every $ρ\in [0,1]$, the \emph{$d$-constrained two-terminal reliability of $G$ at $ρ$}, denoted $R_G^d(ρ)$, is the probability that $G$ has some path of length at most $d$ joining its terminals after each of its edges is independently deleted with probability $ρ$. We say $G$ is a \emph{$d$-uniformly most reliable two-terminal graph} ($d$-UMRTTG) if for each $H$ in $T_{n,m}$ and every $ρ\in [0,1]$ it holds that $R_{G}^d(ρ)\geq R_{H}^{d}(ρ)$. Previous works studied the existence of $d$-UMRTTG in $T_{n,m}$ when $d$ is greater than or equal to $n-1$, or equivalently, when the distance constraint is dropped. In this work, a characterization of all $1$-UMRTTGs and $2$-UMRTTGs is given. Several structural properties of 3-UMRTTGs are established. For every $d\geq 3$, we prove the existence of a unique $d$-UMRTTG in $T_{n,\binom{n}{2}-1}$. In contrast, for every $d\geq 4$ and every $n\geq 11$, no $d$-UMRTTG exists in $T_{n,m}$ whenever $20\leq m\leq 3n-9$ or $3n-5\leq m\leq \binom{n}{2}-2$. Additionally, for each $d\geq 5$, it is proved that no $d$-UMRTTG exists in $T_{n,m}$ when $n\geq 11$ and $m=3n-8$.