In this paper, we introduce a subclass of chordal graphs which contains $d$-trees and show that their complement are vertex decomposable and so is shellable and sequentially Cohen-Macaulay.

The classes of sequentially Cohen-Macaulay and sequentially homotopy Cohen-Macaulay complexes and posets are studied. First, some different versions of the definitions are discussed and the homotopy type is determined. Second, it is shown how various constructions, such as join, product and rank-selection preserve these properties. Third, a characterization of sequential Cohen-Macaulayness for ...

Let I ⊂ R be a graded ideal in the polynomial ring R = K[x1, . . . , xn] where K is a field, and fix a term order <. It has been shown in [17] that the Hilbert functions of the local cohomology modules of R/I are bounded by those of R/ in(I), where in(I) denotes the initial ideal of I with respect to <. In this note we study the question when the local cohomology modules of R/I and R/ in(I) hav...

We prove that sequentially Cohen-Macaulay rings in positive characteristic, as well as sequentially Cohen-Macaulay Stanley-Reisner rings in any characteristic, have trivial Lyubeznik table. Some other configurations of Lyubeznik tables are also provided depending on the deficiency modules of the ring.

Let G be a simple (i.e., no loops and no multiple edges) graph. We investigate the question of how to modify G combinatorially to obtain a sequentially CohenMacaulay graph. We focus on modifications given by adding configurations of whiskers to G, where to add a whisker one adds a new vertex and an edge connecting this vertex to an existing vertex in G. We give various sufficient conditions and...

Björner and Wachs generalized the definition of shellability by dropping the assumption of purity; they also introduced the h-triangle, a doubly-indexed generalization of the h-vector which is combinatorially significant for nonpure shellable complexes. Stanley subsequently defined a nonpure simplicial complex to be sequentially Cohen-Macaulay if it satisfies algebraic conditions that generaliz...