
> 
> **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 $n > 0$.



Properties of very well covered graph
Is the following graph not a counter example?


Consider a graph on 6 vertices $u$, $v$, $w$, $u'$, $v'$, $w'$ with the following structure: $i)$ $u, v, w$ form a triangle, and $ii)$ $u'$ is connected to $u$, $v'$ to $v$ and $w'$ to $w$. 


Each maximal independent set appears to contain $3=|V|/2$ vertices, and $G$ contains exactly one induced odd cycle.


