  We focus our attention on well-covered graphs that are vertex decomposable.
We show that for many known families of these vertex decomposable graphs, the
set of shedding vertices forms a dominating set. We then construct three new
infinite families of well-covered graphs, none of which have this property. We
use these results to provide a minimal counterexample to a conjecture of
Villarreal regarding Cohen-Macaulay graphs.
