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.

A tree T is arbitrarily vertex decomposable if for any sequence of positive integers adding up to the order of T there is a sequence of vertex-disjoint subtrees of T whose orders are given by . An on-line version of the problem of characterizing arbitrarily vertex decomposable trees is completely solved here. © 2007 Elsevier B.V. All rights reserved.

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 ver...

Let G be a graph of order n and r , 1 ≤ r ≤ n, a fixed integer. G is said to be r -vertex decomposable if for each sequence (n1, . . . , nr ) of positive integers such that n1 + · · · + nr = n there exists a partition (V1, . . . , Vr ) of the vertex set of G such that for each i ∈ {1, . . . , r}, Vi induces a connected subgraph of G on ni vertices. G is called arbitrarily vertex decomposable if...

In this paper, we characterize the shellable complete $t$-partite graphs. We also show for these types of graphs the concepts vertex decomposable, shellable and sequentially Cohen-Macaulay are equivalent. Furthermore, we give a combinatorial condition for the Cohen-Macaulay complete $t$-partite graphs.

A n-vertex graph is said to be decomposable if for any partition (λ1, . . . , λp) of the integer n, there exists a sequence (V1, . . . , Vp) of connected vertex-disjoint subgraphs with |Vi| = λi. The aim of the paper is to study the homeomorphism classes of decomposable trees. More precisely, we show that homeomorphism classes containing decomposable trees with an arbitrarily large minimal dist...

A graph G of order n is said to be arbitrarily vertex decomposable if for each sequence (n1, . . . , nk) of positive integers such that n1 + · · · + nk = n there exists a partition (V1, . . . , Vk) of the vertex set of G such that for each i ∈ {1, . . . , k}, Vi induces a connected subgraph of G on ni vertices. The main result of the paper reads as follows. Suppose that G is a connected graph o...