Definition: Very well covered graph to be a well-covered graph (possibly disconnected, but with no isolated vertices) in which each maximal independent set (and therefore also each minimal vertex cover) contains exactly half of the vertices.
Let $G$ be a very well covered graph.
Prove\disprove: Either $G$ does not have any induced odd cycle or $G$ have $2n$ number of induced odd cycles, where $n > 0$.