SUMMARY
Let G=(V,E)G=(V,E) be a graph. If GG is a König graph or if GG is a graph without 3-cycles and 5-cycles, we prove that the following conditions are equivalent: ?G\Delta_{G} is pure shellable, R/I?R/I_{\Delta} is Cohen-Macaulay, GG is unmixed vertex decomposable graph and GG is well-covered with a perfect matching of König type e1,…,ege_{1},\dots,e_{g} without 4-cycles with two eie_i'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.