scholarly journals Connected and Total Vertex covering in Graphs

2021 ◽  
Vol 12 (2) ◽  
Author(s):  
Sathikala L, Et. al.
Keyword(s):  
Vertex Cover ◽  
Covering Number ◽  
Induced Subgraph ◽  
Minimum Cardinality ◽  
End Point ◽  
Vertex Covering ◽  
Connected Vertex Cover ◽  
Connected Vertex

A Subset S of vertices of a Graph G is called a vertex cover if S includes at least one end point of every edge of the Graph. A Vertex cover S of G is a connected vertex cover if the induced subgraph of S is connected. The minimum cardinality of such a set is called the connected vertex covering number and it is denoted by    . A Vertex cover S of G is a total vertex cover if the induced subgraph of S has no isolates. The minimum cardinality of such a set is called the total vertex covering number and it is denoted by  .In this paper a few properties of connected vertex cover and total vertex covers are studied and specific values of   and   of some well-known graphs are evaluated.  

Secure vertex cover of a graph

2017 ◽  
Vol 09 (02) ◽  
pp. 1750026 ◽  
Author(s):  
P. Roushini Leely Pushpam ◽  
Chitra Suseendran
Keyword(s):  
Vertex Cover ◽  
Covering Number ◽  
Minimum Cardinality ◽  
Protection Strategy ◽  
Vertex Covering

We study the problem of using mobile guards to defend the vertices of a graph [Formula: see text] against a single attack on its vertices. A vertex cover of a graph [Formula: see text] is a set [Formula: see text] such that for each edge [Formula: see text], at least one of [Formula: see text] or [Formula: see text] is in [Formula: see text]. The minimum cardinality of a vertex cover is termed the vertex covering number and it is denoted by [Formula: see text]. In this context, we introduce a new protection strategy called the secure vertex cover[Formula: see text]SVC[Formula: see text] problem, where the guards are placed at the vertices of a graph, in order to protect the graph against a single attack on its vertices. We are concerned with the protection of [Formula: see text] against a single attack, using at most one guard per vertex and require the set of guarded vertices to be a vertex cover. In addition, if any guard moves along an edge to deal with an attack to an unguarded vertex, then the resulting placement of guards must also form a vertex cover. Formally, this protection strategy defends the vertices of a graph against a single attack and simultaneously protects the edges. We define a SVC to be a set [Formula: see text] such that [Formula: see text] is a vertex cover and for each [Formula: see text], there exists a [Formula: see text] such that [Formula: see text] is a vertex cover. The minimum cardinality of an SVC is called the secure vertex covering number and it is denoted by [Formula: see text]. In this paper, a few properties of SVC of a graph are studied and specific values of [Formula: see text] for few classes of well-known graphs are evaluated.

scholarly journals Extension and Its Price for the Connected Vertex Cover Problem

2019 ◽  
pp. 315-326
Author(s):  
Mehdi Khosravian Ghadikoalei ◽  
Nikolaos Melissinos ◽  
Jérôme Monnot ◽  
Aris Pagourtzis
Keyword(s):  
Vertex Cover ◽  
Vertex Cover Problem ◽  
Connected Vertex Cover ◽  
Cover Problem ◽  
Connected Vertex

Enumerate and Expand: Improved Algorithms for Connected Vertex Cover and Tree Cover

2007 ◽  
Vol 43 (2) ◽  
pp. 234-253 ◽  
Author(s):  
Daniel Mölle ◽  
Stefan Richter ◽  
Peter Rossmanith
Keyword(s):  
Vertex Cover ◽  
Tree Cover ◽  
Connected Vertex Cover ◽  
Connected Vertex

scholarly journals A PTAS for the minimum weight connected vertex cover P3 problem on unit disk graphs

2015 ◽  
Vol 571 ◽  
pp. 58-66 ◽  
Author(s):  
Limin Wang ◽  
Xiaoyan Zhang ◽  
Zhao Zhang ◽  
Hajo Broersma
Keyword(s):  
Unit Disk ◽  
Minimum Weight ◽  
Vertex Cover ◽  
Unit Disk Graphs ◽  
Connected Vertex Cover ◽  
Connected Vertex

scholarly journals On the Approximate Compressibility of Connected Vertex Cover

Algorithmica ◽  
2020 ◽  
Vol 82 (10) ◽  
pp. 2902-2926
Author(s):  
Diptapriyo Majumdar ◽  
M. S. Ramanujan ◽  
Saket Saurabh
Keyword(s):  
Vertex Cover ◽  
Connected Vertex Cover ◽  
Connected Vertex
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