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.