… A graph G is called r-regular if degG(v) = r, for each vertex v of G. Let G1 and G2 be two …
families of well-covered graphs and vertex decomposable graphs are given. Now we are ready …

… Planar regular 1 -well-covered graphs are studied in [32]. 1 -… and vertex decomposability,
in particular. In addition, we show that for a well-covered graph G without isolated vertices, G is …

… the regularity of these ideals. More explicit, we prove that for a bipartite graph G and every
integer k ≥ 1, the regularity of S/J(G)k is at … We are now ready to define vertex decomposable …

… ) G is a bipartite graph and unmixed; (iv) G is a bipartite graph and … is a very well-covered
graph, where the graph G is called very well-covered if it is unmixed without an isolated vertices …

… is a Cohen–Macaulay and very well-covered graph, then J(G) … –Mumford regularity of symbolic
powers of certain vertex cover … section is to prove that Gk is a vertex-decomposable graph …

… vertex-decomposable graphs [2]. Furthermore, this notion allows us to provide another
characterization of W2 … A graph is called k-regular if every vertex in the graph is of degree k. A leaf …

… In this paper we study two interrelated graph classes, the class of well-covered graphs
and … , of other well-studied objects in graphs; for example, transforming a regular triangle-free …

… cases in which we find a shellable or Cohen–Macaulay circulant graph, it is also vertex
decomposable. Our new example of a graph that is shellable but not vertex decomposable, ie, …

… invariants in well-covered forests, well-covered chordal graphs, Cohen–… is a subclass of
well-covered graphs [14, Theorem 3.1] and every graph in this class is vertex decomposable [4, …

… ideal is componentwise linear then its regularity is equal to … ] that G is a Cohen-Macaulay
very well-covered graph if and only … vertex decomposable if its independence complex is vertex …