Paper Info
Title
Vertex-decomposable Graphs, Codismantlability, Cohen-Macaulayness, and Castelnuovo-Mumford Regularity.
Abstract
We call a vertex x of a graph G = (V, E) a codominated vertex if N-G[y] subset of N-G[x] for some vertex y is an element of 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 (C-4, C-5)-free vertex-decomposable graphs are codismantlable, and prove that if G is a (C-4, C-5, C-7)-free well-covered graph, then vertex-decomposability, 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 (C-4, C-5)-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 H-n (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(H-n) = im(H-n) if n >= 6. This answers a recent question of Mahmoudi, et al [12].
Year
Venue
Keywords
2014
ELECTRONIC JOURNAL OF COMBINATORICS
Cohen-Macaulay and sequentially Cohen-Macaulay graphs,vertex decomposable graphs,well-covered graphs,codismantlability,induced matching,co-chordal cover number,edge rings,Castelnuovo-Mumford regularity
Field
DocType
Volume
Discrete mathematics,Combinatorics,Indifference graph,Vertex (geometry),Interval graph,Vertex (graph theory),Chordal graph,Bipartite graph,Neighbourhood (graph theory),Pathwidth,Mathematics
Journal
21.0
Issue
ISSN
Citations 
1.0
1077-8926
2
PageRank 
References 
Authors
0.43
6
2
Name
Order
Citations
PageRank
Türker Bíyíkoglu1887.40
Yusuf Civan272.73