  We call a (simple) graph G codismantlable if either it has no edges or else
it has a codominated vertex x, meaning that the closed neighborhood of x
contains that of one of its neighbor, such that G-x codismantlable. We prove
that if G is well-covered and it lacks induced cycles of length four, five and
seven, than the vertex decomposability, codismantlability and
Cohen-Macaulayness for G are all equivalent. The rest deals with the
computation of Castelnuovo-Mumford regularity of codismantlable graphs. Note
that our approach complements and unifies many of the earlier results on
bipartite, chordal and very well-covered graphs.
