Existence of most reliable two-terminal graphs with distance constraints

TL;DR

The paper investigates the existence of distance-constrained uniformly most reliable two-terminal graphs ($d$-UMRTTGs) within the class $T_{n,m}$. It introduces $d$-pathsets and the reliability polynomial $R_G^d(\rho)$, along with the notions of $d$-stronger and $d$-LMRTTG to compare graphs. It fully characterizes 1- and 2-UMRTTGs, proving a unique 3-UMRTTG $H_{n,m}$ for $n\ge 6$ and $5\le m\le 2n-3$, and establishes nonexistence of $d$-UMRTTGs for all $d\ge 4$ in several $(n,m)$ ranges (e.g., $n\ge 11$, $20\le m \le 3n-9$ or $3n-5\le m \le {n\choose 2}-2$). The results extend distance-unconstrained findings and reveal structural constraints (via irrelevance, universality, and almost-regularity) that limit higher-distance UMRTTGs. These insights advance understanding of reliability-optimized network designs under distance constraints.

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 nonisomorphic 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 each $ρ\in [0,1]$, 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. Then, it is proved that there exists a unique $3$-UMRTTG in $T_{n,m}$ when $n\geq 6$ and $5 \leq m \leq 2n-3$. Finally, for each $d\geq 4$ and each $n\geq 11$ it is proved that there is no $d$-UMRTTG in $T_{n,m}$ when $20 \leq m \leq 3n-9$ or when $3n-5 \leq m \leq \binom{n}{2}-2$.

Figures (1)

• Figure 1: Two-terminal graph $A_{n,r}$

Theorems & Definitions (50)

• Definition 1
• Definition 2
• Definition 3
• Remark 4
• Lemma 5
• proof
• Remark 6
• Definition 7
• Definition 8
• Lemma 9: Bertrand et al. 2018Bertrand