On the most reliable graphs with fixed redundancy

2026-03-01

AI Summary

This study investigates the construction of graphs with a fixed number of vertices \(n\) and edges \(m\)—equivalently, fixed redundancy \(r = m - (n-1)\)—that maximize all-terminal reliability under an edge-independent failure model for all failure probabilities \(p \in [0,1]\). Focusing on sparse graphs, the work characterizes optimal graph properties through analyses of minimum cut-set structure, girth, regularity, and the local behavior of reliability polynomials. Key contributions include partial confirmation of Bourel et al.'s conjecture that high-girth regular graphs are optimal, a novel construction method based on subdivisions of specific cubic graphs, identification of the unique candidate cubic graph for redundancies \(r \leq 19\), proof that uniformly most reliable graphs do not exist for certain redundancies as \(n\) grows large, and the development of a general construction framework applicable to large \(n\) and moderate redundancy levels.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Web data provenance, reliability, and authenticity

Abstract

The all-terminal reliability of a graph $G$ is the probability that $G$ remains connected when each edge fails independently with probability $p$. For fixed $n$ and $m$, the uniformly most reliable problem asks which graph with $n$ vertices and $m$ edges maximizes reliability for all $p \in [0,1]$. Although such graphs do not always exist, optimal graphs in the regime $p \to 0$ always do and are determined by the structure of their minimal cut sets. We establish a structural characterization of graphs that are most reliable near $p=0$. Our results partially resolve a conjecture of Bourel et al., showing that, under suitable conditions, regular graphs with maximal girth are optimal. Extending this analysis to graphs with fixed redundancy $r=m-(n-1)$ and sufficiently large $n$, we show that the most reliable graphs are obtained by subdividing the most reliable cubic graphs with $2(r-1)$ vertices. The general conjecture remains open. Unlike previous results, which resolved only small redundancy cases or very dense regimes, our approach yields a substantial extension of the known range. We determine the unique cubic candidates for uniformly most reliable graphs for all redundancy levels $m-n \le 19$, and prove the non-existence of uniformly most reliable graphs for several infinite families with fixed redundancy and asymptotically large $n$. These results significantly enlarge both the candidate class and the range of provable non-existence.

Problem

• all-terminal reliability
• uniformly most reliable graph
• graph redundancy
• minimal cut sets
• cubic graphs

Innovation

• all-terminal reliability
• uniformly most reliable graph
• fixed redundancy
• cubic graph
• girth