A new view toward vertex decomposable graphs

Abstract

In this paper, we bring a new view about closed neighbourhood to show the vertex decomposability of graphs. Making use of the characterization of hereditary vertex decomposable graphs, we introduce a class of vertex decomposable graphs, which include some well-known classic vertex decomposable graphs such as clique-whiskered graphs and Cameron-Walker graphs.

Introduction

As we know, vertex decomposable graphs are an important class of algebraic graphs, since vertex decomposable graphs are shellable, and well-covered shellable graphs are Cohen-Macaulay. It is known that chordal graphs form an important class of vertex decomposable graphs, this is characterized by R. Woodroofe in [14]. D. Cook and N. Nagel give a method to construct Cohen-Macaulay graphs in [2], actually, they construct a class of vertex decomposable graphs, called clique-whiskered graphs. In [6], T. Hibi et al. prove that Cameron-Walker graphs are vertex decomposable, and hence they are sequentially Cohen-Macaulay. Recently, Liu and Wu prove that Boolean graphs are well-covered and vertex decomposable in [8], which shows that Boolean graphs are a new class of Cohen-Macaulay graphs.

Recall that a simplicial complex (abbreviated as complex) Δ on a vertex set V is a finite subset of $2^V$, such that $A\in \mathrm{\Delta }$ and $B\subseteq A$ implies $B\in \mathrm{\Delta }$. The set A is called a face if $A\in \mathrm{\Delta }$, and called a facet if A is a maximal face with respect to inclusion. The set of facets of Δ is denoted by $\mathcal{F}(\mathrm{\Delta })$. The dimension of a face A, denoted by $\dim (A)$, is $|A|-1$. The dimension of a complex Δ, denoted by $\dim (\mathrm{\Delta })$, is the maximum dimension of its faces. The complex Δ is called pure if all the facets of Δ have the same dimension; otherwise, it is called nonpure. For a vertex $x\in V$, the link and the deletion of x in Δ is defined by

$$lin{k}_{\mathrm{\Delta }}(x)=〈F|x\notin F,F\cup \{x\}\in \mathcal{F}(\mathrm{\Delta })〉,\mathrm{\Delta }\setminus x=〈F|x\notin F\in \mathrm{\Delta }〉.$$

A simplicial complex Δ is vertex decomposable if either Δ is a simplex (a complex with only one facet), or there exists a vertex x of Δ such that

(1) $lin{k}_{\mathrm{\Delta }}(x)$ and $\mathrm{\Delta }\setminus x$ are vertex decomposable, and

(2) Each facet of $lin{k}_{\mathrm{\Delta }}(x)$ is not a facet of $\mathrm{\Delta }\setminus x$.

A vertex x satisfying conditions (1) and (2) is called a shedding vertex of Δ. If x only satisfies condition (2), then it is called a weak shedding vertex.

A simplicial complex Δ is called shellable if there exists a linear order $F_1,\cdots ,F_m$ on its facet set $\mathcal{F}(\mathrm{\Delta })$ such that for each pair $i<j$, there exists a $k<j$, such that $|F_j\setminus F_k|=1$ and $F_i\cap F_j\subseteq F_k$. Correspondingly, if we require further that $F_k\subseteq F_i\cup F_j$, then Δ is called strongly shellable. Note that on pure complexes, there are the following deductive relationships:

pure vertex decomposability ⟹ pure shellability ⟹ Cohen-Macaulayness.

For other concepts and results about simplicial complex, please refer to [5].

Let G be a simple graph. Denote by $V(G)$ and $E(G)$ its vertex set and edge set respectively. A subset T of $V(G)$ is called an independent set if each pair of vertices in T are nonadjacent in G. An independent set is called maximal if it is maximal with respect to inclusion. A graph is well-covered if all maximal independent sets have the same cardinality.

A simplicial complex Δ is called the independence complex of a graph G, denoted by $\mathrm{\Delta }(G)$, if the facet set $\mathcal{F}(\mathrm{\Delta })$ consists of all maximal independent sets of the graph G. It is clear that $\mathrm{\Delta }(G)$ is pure if and only if G is well-covered. For a graph G, it is said to be vertex decomposable (shellable, respectively), if its independence complex $\mathrm{\Delta }(G)$ is vertex decomposable (shellable, respectively). Now assume further G has vertex set $V(G)=\{x_1,\dots ,x_n\}$ with edge set $E(G)$, and let $R=K[x_1,x_2,\dots ,x_n]$ be the polynomial ring with n variables $x_1,\dots ,x_n$ over a field K. Recall that the edge ideal associated to G is defined to be the ideal $I(G)=〈x_i{x}_j|\{x_i,x_j\}\in E(G)〉$ of R, and it is well known that $I(G)$ is identical with the Stanley-Reisner ideal $I_{\mathrm{\Delta }}$ of the simplicial complex $\mathrm{\Delta }=\mathrm{\Delta }(G)$. The edge ideal of a graph was introduced and the Cohen-Macaulay property of the edge ideal was studied by Villarreal, see [13] for details.

This paper is organized as follows: In chapter 2, we introduce a view to see vertex decomposability of a graph by closed neighbourhoods. Then, hereditary vertex decomposable graphs are characterized in chapter 3. Making use of the view about closed neighbourhood, a class of vertex decomposable graphs which are called core vertex decomposable graphs are introduced in chapter 4. Finally, in chapter 5, a class of well-covered vertex decomposable graphs, which are actually clique-whiskered graphs, are restudied.