The bunkbed conjecture for graph percolation

Let $G$ be a graph, and let $BB(G)$ be its bunkbed graph with two layers, whose corresponding vertices are denoted $x^-$ and $x^+$ for $x obreak\text{ in } obreak V(G)$. Let

$$\mu \colon E(BB(G)) \to [0,1]$$

be a symmetric weight on $BB(G)$, and write $\mathbb{P}_{BB(G),\mu}(u\sim v)$ for the probability that $u$ and $v$ are connected after independently retaining each edge according to its weight. Bunkbed conjecture. For every $x,y\in V(G)$,

$$\mathbb{P}_{BB(G),\mu}(x^-\sim y^-)\geq \mathbb{P}_{BB(G),\mu}(x^-\sim y^+).$$

The conjecture asserts that, under symmetric edge percolation, vertices in the same layer are at least as likely to be connected as vertices in opposite layers. The paper proves that the conjecture is preserved when graphs are glued along a vertex, with forests as an immediate consequence; explicit proven families include complete graphs, complete bipartite graphs, and complete graphs minus one edge.