Dense Arbitrarily Vertex Decomposable Graphs

Constructing vertex decomposable graphs

‎Recently‎, ‎some techniques such as adding whiskers and attaching graphs to vertices of a given graph‎, ‎have been proposed for constructing a new vertex decomposable graph‎. ‎In this paper‎, ‎we present a new method for constructing vertex decomposable graphs‎. ‎Then we use this construction to generalize the result due to Cook and Nagel‎.

Recursively Arbitrarily Vertex-decomposable Suns

A graph G = (V,E) is arbitrarily vertex decomposable if for any sequence τ of positive integers adding up to |V |, there is a sequence of vertex-disjoint subsets of V whose orders are given by τ , and which induce connected graphs. The aim of this paper is to study the recursive version of this problem on a special class of graphs called suns. This paper is a complement of [O. Baudon, F. Gilber...

Bounds for the Regularity of Edge Ideal of Vertex Decomposable and Shellable Graphs

An Ore-type condition for arbitrarily vertex decomposable graphs

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...

Complement of Special Chordal Graphs and Vertex Decomposability

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.