… 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 …

… ) 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 …

… 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 …

… , shellability and vertex-decomposability of very well-covered graphs as well as their
equivalency. In Sect. 3, we deal with regularity of the edge ideals of these graphs. In Sect. 4, a …

… regularity of I ( G ) q and we prove the conjectured upper bound for the class of vertex
decomposable graphs. … While in [30], the hypothesis was that the graph is very well-covered, it …

… show that if G is a unicyclic vertex decomposable graph, then … if G is a Cohen-Macaulay
and very well covered graph, then … find the regularity of symbolic powers of vertex cover ideals …

… We prove that if G is a very well-covered graph then for all s ≥ 1 the regularity of I(G)s is …
In [13], Gitler and Valencia proved that if G is a well-covered graph without isolated vertices, …

… Our goal is to show that the jets of very well-covered graphs are very well-covered, which
was conjectured in [GHW21, … Vertex decomposability and regularity of very wellcovered …