On well covered, vertex decomposable and Cohen - Macaulay graphs (2016)
LetG= (V, E) be a graph. IfGis a K ̈onig graph or ifGis a graph without3-cycles and 5-cycles, we prove that the following conditions are equivalent: ∆Gis pure shellable,R/I∆is Cohen-Macaulay,Gis an unmixed vertex decomposablegraph andGis well-covered with a perfect atching of K ̈onig typee1, . . . , egwithout4-cycles with twoei’s. Furthermore, we study vertex decomposable and shellable(non-pure) properties in raphs without 3-cycles and 5-cycles. Finally, we give someproperties and relations between critical, extendable and shedding vertices.