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.Comment: 11 page