Petersen graph vertex-irreducibility conjecture

A bridgeless cubic class 2 graph $G$ is vertex-irreducible if, for any two vertices $x,y\in V(G)$, the graph $G-\{x,y\}$ cannot be extended to a bridgeless cubic class 2 graph by adding edges.

Petersen graph vertex-irreducibility conjecture. The Petersen graph is the only vertex-irreducible bridgeless cubic class 2 graph.

A vertex-irreducible cubic graph is known to be cyclically 5-edge-connected, and the conjecture was verified computationally for all cubic bridgeless graphs with at most 36 vertices. The general statement remains open in the source.