  A graph $G$ is well-covered if it has no isolated vertices and all the
maximal independent sets have the same cardinality. If furthermore two times
this cardinality is equal to $|V(G)|$, the graph $G$ is called very
well-covered. The class of very well-covered graphs contains bipartite
well-covered graphs. Recently in \cite{CRT} it is shown that a very
well-covered graph $G$ is Cohen-Macaulay if and only if it is pure shellable.
In this article we improve this result by showing that $G$ is Cohen-Macaulay if
and only if it is pure vertex decomposable. In addition, if $I(G)$ denotes the
edge ideal of $G$, we show that the Castelnuovo-Mumford regularity of $R/I(G)$
is equal to the maximum number of pairwise 3-disjoint edges of $G$. This
improves Kummini's result on unmixed bipartite graphs.
