  Let $G=(V,E)$ be a graph. If $G$ is a K\"onig graph or $G$ is a graph without
3-cycles and 5-cycle, we prove that the following conditions are equivalent:
$\Delta_{G}$ is pure shellable, $R/I_{\Delta}$ is Cohen-Macaulay, $G$ is
unmixed vertex decomposable graph and $G$ is well-covered with a perfect
matching of K\"onig type $e_{1},...,e_{g}$ without square with two $e_i$'s. We
characterize well-covered graphs without 3-cycles, 5-cycles and 7-cycles. Also,
we study when graphs without 3-cycles and 5-cycles are vertex decomposable or
shellable. Furthermore, we give some properties and relations between critical,
extendables and shedding vertices. Finally, we characterize unicyclic graphs
with each one of the following properties: unmixed, vertex decomposable,
shellable and Cohen-Macaulay.
