The edge geodetic vertex covering number of a graph

Abstract A confluent and terminating reduction system is introduced for graphs, which preserves the number of their perfect matchings. A union-find algorithm is presented to carry out reduction in almost linear time. The König property is investigated in the context of reduction by introducing the König deficiency of a graph G as the difference between the vertex covering number and the matching number of G. It is shown that the problem of finding the König deficiency of a graph is NP-complete even if we know that the graph reduces to the empty graph. Finally, the König deficiency of graphs G having a vertex v such that $$G-v$$G-v has a unique perfect matching is studied in connection with reduction.

Download Full-text

Secure vertex cover of a graph

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830917500264 ◽

2017 ◽

Vol 09 (02) ◽

pp. 1750026 ◽

Cited By ~ 1

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.

Download Full-text

Inverse vertex covering number of a graph

Journal of Discrete Mathematical Sciences and Cryptography ◽

10.1080/09720529.2012.10698391 ◽

2012 ◽

Vol 15 (6) ◽

pp. 389-393 ◽

Cited By ~ 3

Author(s):

V. R. Kulli ◽

R. R. Iyer

Keyword(s):

Covering Number ◽

Vertex Covering

Download Full-text

Visual Cryptography Scheme for Graphs with Vertex Covering Number Two

National Academy Science Letters ◽

10.1007/s40009-013-0171-x ◽

2013 ◽

Vol 36 (6) ◽

pp. 575-579 ◽

Cited By ~ 1

Author(s):

S. Arumugam ◽

R. Lakshmanan ◽

Atulya K. Nagar

Keyword(s):

Visual Cryptography ◽

Covering Number ◽

Vertex Covering ◽

Visual Cryptography Scheme

Download Full-text

Connected and Total Vertex covering in Graphs

Turkish Journal of Computer and Mathematics Education (TURCOMAT) ◽

10.17762/turcomat.v12i2.1901 ◽

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.

Download Full-text

Vertex cover and Edge vertex domination in trees

Proyecciones (Antofagasta) ◽

10.22199/issn.0717-6279-3532 ◽

2021 ◽

Vol 40 (5) ◽

pp. 1147-1154

Author(s):

B. Senthilkumar ◽

H. Naresh Kumar ◽

Y. B. Venkatakrishnan

Keyword(s):

Dominating Set ◽

Vertex Cover ◽

Domination Number ◽

Simple Graph ◽

Covering Number ◽

Vertex Covering

Let G = (V,E) be a simple graph. An edge e ∈ E(G) edge-vertex dominates a vertex v ∈ V (G) if e is incident with v or e is incident with a vertex adjacent to v. A subset D ⊆ E(G) is an edge-vertex dominating set of a graph G if every vertex of G is edge-vertex dominated by an edge of D. A vertex cover of G is a set C ⊆ V such that for each edge uv ∈ E at least one of u and v is in C. We characterize trees with edge-vertex domination number equals vertex covering number.

Download Full-text

m-Secure vertex cover of a graph

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830918500751 ◽

2018 ◽

Vol 10 (06) ◽

pp. 1850075

Author(s):

P. Roushini Leely Pushpam ◽

Chitra Suseendran

Keyword(s):

Vertex Cover ◽

Covering Number ◽

The Other ◽

Vertex Cover Problem ◽

Vertex Covering ◽

Before And After ◽

Minimum Number ◽

Cover Problem

In this paper, we use multiple guard movements to defend the edges of a graph [Formula: see text] against a single attack. At most, one guard is positioned at each vertex. To defend an attack on an edge, a guard at an incident vertex moves across the attacked edge and the other guards may move (or not) to the neighboring vertices to better configure themselves. This strategy requires the set of vertices containing guards to be a vertex cover before and after an attack. A suitable placement of guards is called an [Formula: see text]-secure vertex cover of [Formula: see text]. We call this the [Formula: see text]-secure vertex cover problem, where [Formula: see text] stands for the multiple guard movements. The minimum number of guards required to defend the edges of [Formula: see text] against a single attack using multiple guard movements is called the [Formula: see text]-secure vertex covering number and it is denoted by [Formula: see text]. In this paper we initiate a study of this parameter.

Download Full-text