On the most reliable graphs with fixed redundancy

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.

Figures (8)

• Figure 1: An example of a structure graph. Consecutive vertices with a degree of two form a chain, while vertices with a degree other than two become structure vertices. In optimal graphs, the lengths of the chains differ by at most one. In this scenario, we classify the chains as short or long.
• Figure 2: Various Subgraphs(Skeletons) and their induced cut sets.
• Figure 3: The reliability classes for graphs with $16$ vertices and $24$ edges. All the graphs that are not in the class $\mathcal{A}_5$ contain non-trivial cut sets(red) that are induced from a short cycle or from a bigger subgraph(black).
• Figure 4: Most reliable graphs near zero with a given redundancy $r$, and number of nodes $n=2(r-1)$. $g$ denotes the girth of the graph. Green graphs are Tree-Balanced, and gold graphs are previously proven to be uniformly most reliable.
• Figure 5: Comparison of different chain configurations on the Wagner graph, which is the most reliable graph near zero with $8$ vertices and $12$ edges. In the first and second graphs, the chain lengths differ by at most one. The first graph features long chains forming a matching with the optimal separation vector, making it the more reliable graph near zero. In contrast, the long chains in the second graph do not form a matching. In the third graph, all chains are of equal length except for one very long chain, which maximizes the tree number when removed, and one very short chain, which minimizes the tree number when removed. As predicted, the tree number of the third graph is the highest among the presented graphs.

Theorems & Definitions (38)

• Definition 2.1: Uniformly Most Reliable Graph
• Definition 2.2: Most Reliable Graph Near Zero or One
• Lemma 2.3: Coefficients Comparison brown2014nonexistence
• Proposition 2.4: The Size of the Induced Cut Set
• Theorem 2.5: Regular Graphs Without Non-Trivial Cut Sets
• Lemma 2.6: Extension of Coefficient Comparison
• proof
• Definition 2.7: Marked structure
• Definition 2.8: Asymptotically most reliable marked structure
• Theorem 2.9: Most reliable large graphs near zero