Baker, Jonathan, Kevin N. Vander Meulen, Adam Van Tuyl. “Shedding vertices of vertex decomposable well-covered graphs.” Discrete Mathematics 341, no. 12 (2018): 3355–3369.
Abstract
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.