Vertex-decomposable Graphs, Codismantlability, Cohen-Macaulayness, and Castelnuovo-Mumford Regularity

We call a vertex x of a graph G = (V,E) a codominated vertex if NG[y] ⊆ NG[x] for some vertex y ∈ V \{x}, and a graph G is called codismantlable if either it is an edgeless graph or it contains a codominated vertex x such that G − x is codismantlable. We show that (C4, C5)-free vertex-decomposable graphs are codismantlable, and prove that if G is a (C4, C5, C7)-free well-covered graph, then vertexdecomposability, codismantlability and Cohen-Macaulayness for G are all equivalent. These results complement and unify many of the earlier results on bipartite, chordal and very well-covered graphs. We also study the Castelnuovo-Mumford regularity reg(G) of such graphs, and show that reg(G) = im(G) whenever G is a (C4, C5)-free vertex-decomposable graph, where im(G) is the induced matching number of G. Furthermore, we prove that H must be a codismantlable graph if im(H) = reg(H) = m(H), where m(H) is the matching number of H. We further describe an operation on digraphs that creates a vertex-decomposable and codismantlable graph from any acyclic digraph. By way of application, we provide an infinite family Hn (n > 4) of sequentially Cohen-Macaulay graphs whose vertex cover numbers are half of their orders, while containing no vertex of degree-one such that they are vertex-decomposable, and reg(Hn) = im(Hn) if n > 6. This answers a recent question of Mahmoudi, et al [12].