Characterization of locally most split reliable graphs
Pablo Romero
TL;DR
This paper addresses the problem of identifying locally and uniformly optimal two-terminal graphs with respect to split reliability. It develops a proof strategy based on split-reliability polynomials and a lexicographic optimization over graph-structure counts, culminating in a complete characterization: in every nonempty class $T_{n,m}$, the locally most split reliable graphs are precisely those split-equivalent to the two-terminal balloon graph $G_{n,m}$ (balloon with terminals at distance equal to its diameter). It also proves there is no uniformly most split reliable graph in a broad range of parameters, by constructing counterexamples that outperform the balloon-based candidate in $N_{n-2}$ for large $n$ and $m$. The results unify balloon-graph properties with split reliability and clarify when a global optimum cannot exist, informing both theory and potential applications in reliable message dissemination. Overall, the work advances understanding of split reliability structure and its interplay with fundamental graph invariants like bridges, edge-connectivity, and spanning trees.
Abstract
A two-terminal graph is a graph equipped with two distinguished vertices, called terminals. Let $T_{n,m}$ be the set of all nonisomorphic connected simple two-terminal graphs on $n$ vertices and $m$ edges. Let $G$ be any two-terminal graph in $T_{n,m}$. For every number $p$ in $[0,1]$ we let each of the edges in $G$ be independently deleted with probability $1-p$. The split reliability $SR_{G}(p)$ is the probability that the resulting spanning subgraph has precisely $2$ connected components, each one including one terminal. The two-terminal graph $G$ is uniformly most split reliable if $SR_G(p)\geq SR_{H}(p)$ for each $H$ in $T_{n,m}$ and every $p$ in $[0,1]$. We say $G$ is locally most split reliable if there exists $δ>0$ such that $SR_G(p)\geq SR_{H}(p)$ for each $H$ in $T_{n,m}$ and every $p$ in $(1-δ,1)$. Brown and McMullin showed that there exists uniformly most split reliable graphs in each class $T_{n,m}$ such that $m=n-1$, $m=\binom{n}{2}$, or $m=\binom{n}{2}-1$. The authors also proved that there is no uniformly most split reliable two-terminal graph in $T_{n,n}$ when $n\geq 6$ and specified in which classes $T_{n,m}$ such that $n\leq 7$ there exist uniformly most split reliable graphs. The existence or nonexistence of uniformly most split reliable graphs in the remaining cases is posed by Brown and McMullin as an open problem. In this work, the set $\mathcal{G}_{n,m}$ consisting of all locally most split reliable graphs is characterized in each nonempty class $T_{n,m}$. It is proved that a graph in $T_{n,m}$ is locally most split reliable if and only if its split reliability equals that of the balloon graph equipped with two terminals whose distance equals its diameter. Finally, it is proved that there is no uniformly most split reliable graph in $T_{n,m}$ when $n\geq 7$ and $n\leq m \leq \binom{n-3}{2}+3$.