scholarly journals OPEN PACKING NUMBER FOR SOME CLASSES OF PERFECT GRAPHS

2020 ◽  
Vol 6 (2) ◽  
pp. 38
Author(s):  
K. Raja Chandrasekar ◽  
S. Saravanakumar
Keyword(s):  
Bipartite Graph ◽  
Perfect Graphs ◽  
Perfect Graph ◽  
Maximum Cardinality ◽  
Common Neighbor ◽  
Packing Number ◽  
Vertex Set ◽  
Open Packing ◽  
Split Graphs

Let \(G\) be a graph with the vertex set \(V(G)\).  A subset \(S\) of \(V(G)\) is an open packing set of \(G\) if every pair of vertices in \(S\) has no common neighbor in \(G.\)  The maximum cardinality of an open packing set of \(G\) is the open packing number of \(G\) and it is denoted by \(\rho^o(G)\).  In this paper, the exact values of the open packing numbers for some classes of perfect graphs, such as split graphs, \(\{P_4, C_4\}\)-free graphs, the complement of a bipartite graph, the trestled graph of a perfect graph are obtained.

Open packing bondage number of a graph

2019 ◽  
Vol 11 (05) ◽  
pp. 1950051
Author(s):  
S. Saravanakumar ◽  
A. Anitha ◽  
I. Sahul Hamid
Keyword(s):  
Maximum Cardinality ◽  
Common Neighbor ◽  
Packing Number ◽  
Bondage Number ◽  
Open Packing

In a graph [Formula: see text], a set [Formula: see text] is said to be an open packing set if no two vertices of [Formula: see text] have a common neighbor in [Formula: see text] The maximum cardinality of an open packing set is called the open packing number and is denoted by [Formula: see text]. The open packing bondage number of a graph [Formula: see text], denoted by [Formula: see text], is the cardinality of the smallest set of edges [Formula: see text] such that [Formula: see text]. In this paper, we initiate a study on this parameter.

Changing and unchanging Open Packing: Edge removal

2016 ◽  
Vol 08 (01) ◽  
pp. 1650016 ◽  
Author(s):  
I. Sahul Hamid ◽  
S. Saravanakumar
Keyword(s):  
Maximum Cardinality ◽  
Common Neighbor ◽  
Packing Number ◽  
Edge Removal ◽  
Open Packing

In a graph [Formula: see text], a nonempty set [Formula: see text] is said to be an open packing set if no two vertices of [Formula: see text] have a common neighbor in [Formula: see text] The maximum cardinality of an open packing set is called the open packing number and is denoted by [Formula: see text]. In this paper, we examine the effect of [Formula: see text] when [Formula: see text] is modified by deleting an edge.

scholarly journals On the Open Packing Number of a Graph

2020 ◽  
Vol 8 (4S4) ◽  
pp. 97-100
Keyword(s):  
Maximum Cardinality ◽  
Common Neighbor ◽  
Unicyclic Graphs ◽  
Packing Number ◽  
Open Packing ◽  
Special Classes

A non-empty set of a graph G is an open packing set of G if no two vertices of S have a common neighbor in G. The maximum cardinality of an open packing set is the open packing number of G and is denoted by . An open packing set of cardinality is a -set of G. In this paper, the classes of trees and unicyclic graphs for which the value of is either 2 or 3 are characterized. Moreover, the exact values of the open packing number for some special classes of graphs have been found.

Open packing saturation number of a graph

2016 ◽  
Vol 10 (02) ◽  
pp. 1750033
Author(s):  
I. Sahul Hamid ◽  
S. Saravanakumar
Keyword(s):  
Maximum Cardinality ◽  
Common Neighbor ◽  
Open Packing ◽  
Saturation Number

In a graph [Formula: see text], a non-empty set [Formula: see text] is said to be an open packing set if no two vertices of [Formula: see text] have a common neighbor in [Formula: see text] Let [Formula: see text] and let [Formula: see text] denote the maximum cardinality of an open packing set in [Formula: see text] which contains [Formula: see text]. Then [Formula: see text] is called the open packing saturation number of [Formula: see text]. In this paper, we initiate a study on this parameter.

Outer-connected open packing sets in graphs

2021 ◽  
pp. 2250083
Author(s):  
S. Saravanakumar ◽  
C. Gayathri
Keyword(s):  
Common Neighbor ◽  
Packing Number ◽  
Open Packing

A set [Formula: see text] of a graph [Formula: see text] is an [Formula: see text] [Formula: see text] [Formula: see text] of [Formula: see text] if no two vertices of [Formula: see text] have a common neighbor in [Formula: see text]. An open packing set [Formula: see text] is called an outer-connected open packing set (ocop-set) if either [Formula: see text] or [Formula: see text] is connected. The minimum and maximum cardinalities of an ocop-set are called the lower outer-connected open packing number and the outer-connected open packing number, respectively, and are denoted by [Formula: see text] and [Formula: see text], respectively. In this paper, we initiate a study on these parameters.

HAMILTONIAN PROPERTIES OF FAULTY RECURSIVE CIRCULANT GRAPHS

2002 ◽  
Vol 03 (03n04) ◽  
pp. 273-289 ◽  
Author(s):  
CHANG-HSIUNG TSAI ◽  
JIMMY J. M. TAN ◽  
YEN-CHU CHUANG ◽  
LIH-HSING HSU
Keyword(s):  
Bipartite Graph ◽  
Hamiltonian Path ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Vertex Set ◽  
Regular Node ◽  
Hamiltonian Properties

We present some results concerning hamiltonian properties of recursive circulant graphs in the presence of faulty vertices and/or edges. The recursive circulant graph G(N, d) with d ≥ 2 has vertex set V(G) = {0, 1, …, N - 1} and the edge set E(G) = {(v, w)| ∃ i, 0 ≤ i ≤ ⌈ log d N⌉ - 1, such that v = w + di (mod N)}. When N = cdk where d ≥ 2 and 2 ≤ c ≤ d, G(cdk, d) is regular, node symmetric and can be recursively constructed. G(cdk, d) is a bipartite graph if and only if c is even and d is odd. Let F, the faulty set, be a subset of V(G(cdk, d)) ∪ E(G(cdk, d)). In this paper, we prove that G(cdk, d) - F remains hamiltonian if |F| ≤ deg (G(cdk, d)) - 2 and G(cdk, d) is not bipartite. Moreover, if |F| ≤ deg (G(cdk, d)) - 3 and G(cdk, d) is not a bipartite graph, we prove a more stronger result that for any two vertices u and v in V(G(cdk, d)) - F, there exists a hamiltonian path of G(cdk, d) - F joining u and v.

Graham’s pebbling conjecture holds for the product of a graph and a sufficiently large complete bipartite graph

2019 ◽  
Vol 11 (06) ◽  
pp. 1950068
Author(s):  
Nopparat Pleanmani
Keyword(s):  
Bipartite Graph ◽  
Network Optimization ◽  
Connected Graph ◽  
Complete Bipartite Graph ◽  
Fixed Number ◽  
Arbitrary Vertex ◽  
Graph Pebbling ◽  
Vertex Set ◽  
Graham’S Conjecture ◽  
Pebbling Number

A graph pebbling is a network optimization model for the transmission of consumable resources. A pebbling move on a connected graph [Formula: see text] is the process of removing two pebbles from a vertex and placing one of them on an adjacent vertex after configuration of a fixed number of pebbles on the vertex set of [Formula: see text]. The pebbling number of [Formula: see text], denoted by [Formula: see text], is defined to be the least number of pebbles to guarantee that for any configuration of pebbles on [Formula: see text] and arbitrary vertex [Formula: see text], there is a sequence of pebbling movement that places at least one pebble on [Formula: see text]. For connected graphs [Formula: see text] and [Formula: see text], Graham’s conjecture asserted that [Formula: see text]. In this paper, we show that such conjecture holds when [Formula: see text] is a complete bipartite graph with sufficiently large order in terms of [Formula: see text] and the order of [Formula: see text].

When the annihilator-ideal graph is planar or complete bipartite graph

2019 ◽  
Vol 12 (02) ◽  
pp. 1950024
Author(s):  
M. J. Nikmehr ◽  
S. M. Hosseini
Keyword(s):  
Commutative Ring ◽  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Simple Graph ◽  
Annihilator Ideal ◽  
Vertex Set

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] be the set of ideals of [Formula: see text] with nonzero annihilator. The annihilator-ideal graph of [Formula: see text], denoted by [Formula: see text], is a simple graph with the vertex set [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we present some results on the bipartite, complete bipartite, outer planar and unicyclic of the annihilator-ideal graphs of a commutative ring. Among other results, bipartite annihilator-ideal graphs of rings are characterized. Also, we investigate planarity of the annihilator-ideal graph and classify rings whose annihilator-ideal graph is planar.

scholarly journals VERY WELL-COVERED GRAPHS OF GIRTH AT LEAST FOUR AND LOCAL MAXIMUM STABLE SET GREEDOIDS

2011 ◽  
Vol 03 (02) ◽  
pp. 245-252 ◽  
Author(s):  
VADIM E. LEVIT ◽  
EUGEN MANDRESCU
Keyword(s):  
Perfect Matching ◽  
Local Maximum ◽  
Stable Set ◽  
Stable Sets ◽  
Maximum Cardinality ◽  
Math Program ◽  
The Family ◽  
Maximum Stable Set ◽  
Vertex Set ◽  
Appl Math

A maximum stable set in a graph G is a stable set of maximum cardinality. S is a local maximum stable set of G, and we write S ∈ Ψ(G), if S is a maximum stable set of the subgraph induced by S ∪ N(S), where N(S) is the neighborhood of S. Nemhauser and Trotter Jr. [Vertex packings: structural properties and algorithms, Math. Program.8 (1975) 232–248], proved that any S ∈ Ψ(G) is a subset of a maximum stable set of G. In [Levit and Mandrescu, A new greedoid: the family of local maximum stable sets of a forest, Discrete Appl. Math.124 (2002) 91–101] we have shown that the family Ψ(T) of a forest T forms a greedoid on its vertex set. The cases where G is bipartite, triangle-free, well-covered, while Ψ(G) is a greedoid, were analyzed in [Levit and Mandrescu, Local maximum stable sets in bipartite graphs with uniquely restricted maximum matchings, Discrete Appl. Math.132 (2004) 163–174], [Levit and Mandrescu, Triangle-free graphs with uniquely restricted maximum matchings and their corresponding greedoids, Discrete Appl. Math.155 (2007) 2414–2425], [Levit and Mandrescu, Well-covered graphs and greedoids, Proc. 14th Computing: The Australasian Theory Symp. (CATS2008), Wollongong, NSW, Conferences in Research and Practice in Information Technology, Vol. 77 (2008) 89–94], respectively. In this paper we demonstrate that if G is a very well-covered graph of girth ≥4, then the family Ψ(G) is a greedoid if and only if G has a unique perfect matching.

A UNIFIED APPROACH TO PARALLEL ALGORITHMS FOR THE DOMATIC PARTITION PROBLEM ON SPECIAL CLASSES OF PERFECT GRAPHS

1993 ◽  
Vol 03 (03) ◽  
pp. 233-241
Author(s):  
A. RAMAN ◽  
C. PANDU RANGAN
Keyword(s):  
Parallel Algorithms ◽  
Dominating Set ◽  
Interval Graphs ◽  
Perfect Graphs ◽  
Partition Problem ◽  
Dominating Sets ◽  
Unified Approach ◽  
Block Graphs ◽  
Vertex Set ◽  
Special Classes

A set of vertices D is a dominating set of a graph G=(V, E) if every vertex in V−D is adjacent to at least one vertex in D. The domatic partition of G is the partition of the vertex set V into a maximum number of dominating sets. In this paper, we present efficient parallel algorithms for finding the domatic partition of Interval graphs, Block graphs and K-trees.

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