# $\alpha$-critical graphs and chordless odd cycles

An $\alpha$-critical graph is a graph in which the removal of any edge increases the independence number. Sometimes isolated vertices are forbidden, but that is irrelevant for this question.

It is known that in such a graph every two adjacent edges are on a common chordless odd cycle. I am wondering about the reverse implication. Specifically:

Let $G$ be a graph in which every two adjacent edges are on a common odd chordless cycle. Prove that $G$ is $\alpha$-critical or provide a counterexample.

Btw, I expect it to be false, simply because the book would probably have hinted at an equivalence, if there was one.

It can be verified that every two adjacent edges lie on a common chordless 5-cycle, yet the removal of any edge leaves the independence number at $4=\alpha (P)$. So, the Petersen Graph is not $\alpha$-critical.