The bunkbed conjecture still holds for cactus graphs and for graphs with certain biconnected components

TL;DR

This paper investigates conditions under which the bunkbed conjecture holds, demonstrating it for cactus graphs and certain biconnected components, and analyzing the structure of potential counterexamples.

Contribution

It establishes that the bunkbed conjecture holds for all graphs with biconnected components of specific types, including cactus graphs, and characterizes counterexamples via graph minors.

Findings

Cactus graphs satisfy the strong bunkbed conjecture.

Graphs with certain biconnected components satisfy the weak conjecture.

Counterexamples must contain a subdivision of the diamond graph as a minor.

Abstract

Recently, the bunkbed conjecture has been shown to be false, which naturally prompts questions on how to classify the graphs that still satisfy the conjecture. We distinguish between a weak version of the bunkbed conjecture where all the horizontal edges of the bunkbed graph are present with the same probability and a strong version of the conjecture where the edge weights on the underlying graph may be assigned individually. We show that any given graph satisfies either version of the conjecture if and only if all of its biconnected components do. Moreover, we show that all cactus graphs satisfy the strong version, and by combining previous results of other authors, any graph $G$ such that every biconnected component of $G$ is either a cycle, complete, complete bipartite, symmetric complete $k$-partite or an edge difference of a complete graph and a complete subgraph satisfies the weak version. Furthermore, we apply the aforementioned results to show that any counterexample to the strong version of the bunkbed conjecture contains a non-trivial subdivision of the diamond graph as a minor and demonstrate how this result might be strengthened in the future.