scholarly journals Shellable Cactus Graphs

2010 ◽  
Vol 106 (2) ◽  
pp. 161 ◽  
Author(s):  
Fatemeh Mohammadi ◽  
Dariush Kiani ◽  
Siamak Yassemi
Keyword(s):  
Connected Graphs ◽  
New Class ◽  
Cactus Graphs ◽  
Vertex Decomposable ◽  
Decomposable Graphs

In this paper a new class of vertex decomposable graphs are determined. Moreover, all shellable and sequentially Cohen-Macaulay cactus graphs (i.e., connected graphs in which each edge belongs to at most one cycle) are characterized.

Some families of arbitrarily vertex decomposable graphs

2005 ◽  
Vol 22 ◽  
pp. 451-454 ◽  
Author(s):  
Rafał Kalinowski ◽  
Monika Pilśniak ◽  
Mariusz Woźniak ◽  
Irmina Zioło
Keyword(s):  
Vertex Decomposable ◽  
Decomposable Graphs

scholarly journals Eccentric distance sum index for some classes of connected graphs

2017 ◽  
Vol 71 (2) ◽  
pp. 25 ◽  
Author(s):  
Halina Bielak ◽  
Katarzyna Broniszewska
Keyword(s):  
Lower Bound ◽  
Connected Graphs ◽  
New Class ◽  
Eccentric Distance

In this paper we show some properties of the eccentric distance sum index which is defined as follows \(\xi^{d}(G)=\sum_{v \in V(G)}D(v) \varepsilon(v)\). This index is widely used by chemists and biologists in their researches. We present a lower bound of this index for a new class of graphs.

scholarly journals Recursively arbitrarily vertex-decomposable graphs

2012 ◽  
Vol 32 (4) ◽  
pp. 689 ◽  
Author(s):  
Olivier Baudon ◽  
Frédéric Gilbert ◽  
Mariusz Woźniak
Keyword(s):  
Vertex Decomposable ◽  
Decomposable Graphs

scholarly journals Dense Arbitrarily Vertex Decomposable Graphs

2011 ◽  
Vol 28 (6) ◽  
pp. 807-821 ◽  
Author(s):  
Mirko Horňák ◽  
Antoni Marczyk ◽  
Ingo Schiermeyer ◽  
Mariusz Woźniak
Keyword(s):  
Vertex Decomposable ◽  
Decomposable Graphs

scholarly journals Vertex-Decomposable Graphs, Codismantlability, Cohen-Macaulayness, and Castelnuovo-Mumford Regularity

10.37236/2387 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Türker Bıyıkoğlu ◽  
Yusuf Civan
Keyword(s):  
Vertex Cover ◽  
Infinite Family ◽  
Matching Number ◽  
Induced Matching ◽  
Free Vertex ◽  
Acyclic Digraph ◽  
Vertex Decomposable ◽  
Decomposable Graphs

We call a vertex $x$ of a graph $G=(V,E)$ a codominated vertex if $N_G[y]\subseteq N_G[x]$ for some vertex $y\in V\backslash \{x\}$, 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 $H_n$ ($n\geq 4$) 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 $n\geq 6$. This answers a recent question of Mahmoudi et al.

scholarly journals Symbolic powers of vertex cover ideals

2020 ◽  
Vol 30 (06) ◽  
pp. 1167-1183
Author(s):  
S. Selvaraja
Keyword(s):  
Polynomial Ring ◽  
Vertex Cover ◽  
Simple Graph ◽  
Symbolic Powers ◽  
Vertex Decomposable ◽  
Decomposable Graphs ◽  
Linear Quotients

Let [Formula: see text] be a finite simple graph and [Formula: see text] denote its vertex cover ideal in a polynomial ring over a field [Formula: see text]. In this paper, we show that all symbolic powers of vertex cover ideals of certain vertex-decomposable graphs have linear quotients. Using these results, we give various conditions on a subset [Formula: see text] of the vertices of [Formula: see text] so that all symbolic powers of vertex cover ideals of [Formula: see text], obtained from [Formula: see text] by adding a whisker to each vertex in [Formula: see text], have linear quotients. For instance, if [Formula: see text] is a vertex cover of [Formula: see text], then all symbolic powers of [Formula: see text] have linear quotients. Moreover, we compute the Castelnuovo–Mumford regularity of symbolic powers of certain vertex cover ideals.

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