Independent and vertex covering number on strong product of cycles

2013 ◽  
Vol 7 ◽  
pp. 933-937
Author(s):  
B. Kaewwisetkul ◽  
T. Sitthiwirattham
Keyword(s):  
Covering Number ◽  
Strong Product ◽  
Vertex Covering

scholarly journals The geodetic vertex covering number of a graph

2020 ◽  
Vol 8 (2) ◽  
pp. 683-689
Author(s):  
V.M. Arul Flower Mary ◽  
J. Anne Mary Leema ◽  
P. Titus ◽  
B. Uma Devi
Keyword(s):  
Covering Number ◽  
Vertex Covering

scholarly journals On the König deficiency of zero-reducible graphs

2019 ◽  
Vol 39 (1) ◽  
pp. 273-292
Author(s):  
Miklós Bartha ◽  
Miklós Krész
Keyword(s):  
Perfect Matching ◽  
Linear Time ◽  
Perfect Matchings ◽  
Covering Number ◽  
Matching Number ◽  
Empty Graph ◽  
Reduction System ◽  
Vertex Covering ◽  
Np Complete ◽  
The Difference

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.

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.

Matching and edge covering number on strong product of cycles

2013 ◽  
Vol 7 ◽  
pp. 939-943
Author(s):  
T. Saengngammongkhol ◽  
T. Sitthiwirattham
Keyword(s):  
Covering Number ◽  
Strong Product ◽  
Edge Covering

Visual Cryptography Scheme for Graphs with Vertex Covering Number Two

2013 ◽  
Vol 36 (6) ◽  
pp. 575-579 ◽  
Author(s):  
S. Arumugam ◽  
R. Lakshmanan ◽  
Atulya K. Nagar
Keyword(s):  
Visual Cryptography ◽  
Covering Number ◽  
Vertex Covering ◽  
Visual Cryptography Scheme

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.  

scholarly journals Vertex cover and Edge vertex domination in trees

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.

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