On Well-Covered, Vertex Decomposable and Cohen-Macaulay Graphs

Let G = (V,E) be a graph. If G is a König graph or if G is a graph without 3-cycles and 5-cycles, we prove that the following conditions are equivalent: ∆G is pure shellable, R/I∆ is Cohen-Macaulay, G is an unmixed vertex decomposable graph and G is well-covered with a perfect matching of König type e1, . . . , eg without 4-cycles with two ei’s. Furthermore, we study vertex decomposable and shellable (non-pure) properties in graphs without 3-cycles and 5-cycles. Finally, we give some properties and relations between critical, extendable and shedding vertices.