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 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 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 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.

scholarly journals DOMINATION, TOTAL DOMINATION AND OPEN PACKING OF THE CORCOR DOMAIN OF GRAPHENE

2019 ◽  
pp. 789
Author(s):  
Doost Ali Mojdeh ◽  
Mohammad Habibi ◽  
Leila Badakhshian ◽  
Amir Loghman
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Molecular Graph ◽  
Total Domination ◽  
Total Domination Number ◽  
Common Neighbor ◽  
Open Packing ◽  
New Type ◽  
Carbon Network

A dominating set of a graph G = (V,E) is a subset D of V such that everyvertex not in D is adjacent to at least one vertex in D. A dominating set D is a totaldominating set, if every vertex in V is adjacent to at least one vertex in D. The set Pis said to be an open packing set if no two vertices of P have a common neighbor inG. In this paper, we obtain domination number, total domination number and openpacking number of the molecular graph of a new type of graphene named CorCor thatis a 2-dimensional carbon network.

scholarly journals Cartesian product graphs and k-tuple total domination

Filomat ◽  
2018 ◽  
Vol 32 (19) ◽  
pp. 6713-6731
Author(s):  
Adel Kazemi ◽  
Behnaz Pahlavsay ◽  
Rebecca Stones
Keyword(s):  
Dominating Set ◽  
Cartesian Product ◽  
Minimum Size ◽  
Constructive Proof ◽  
Extremal Case ◽  
Packing Number ◽  
Total Dominating Set ◽  
Product Graphs ◽  
Open Packing ◽  
Cartesian Product Graphs

A k-tuple total dominating set (kTDS) of a graph G is a set S of vertices in which every vertex in G is adjacent to at least k vertices in S; the minimum size of a kTDS is denoted ?xk,t(G). We give a Vizing-like inequality for Cartesian product graphs, namely ?xk,t(G) ?xk,t(H)? 2k?xk,t(G_H) provided ?xk,t(G) ? 2k?(G) holds, where ? denotes the packing number. We also give bounds on ?xk,t(G_H) in terms of (open) packing numbers, and consider the extremal case of ?xk,t(Kn_Km), i.e., the rook?s graph, giving a constructive proof of a general formula for ?x2,t(Kn_Km).

scholarly journals The upper connected vertex detour number of a graph

Filomat ◽  
2012 ◽  
Vol 26 (2) ◽  
pp. 379-388 ◽  
Author(s):  
A.P. Santhakumaran ◽  
P. Titus
Keyword(s):  
Connected Graph ◽  
Proper Subset ◽  
Maximum Cardinality ◽  
Minimum Cardinality ◽  
Positive Integers ◽  
Connected Vertex ◽  
Special Classes

For vertices x and y in a connected graph G = (V, E) of order at least two, the detour distance D(x, y) is the length of the longest x ? y path in G: An x ? y path of length D(x, y) is called an x ? y detour. For any vertex x in G, a set S ? V is an x-detour set of G if each vertex v ? V lies on an x ? y detour for some element y in S: The minimum cardinality of an x-detour set of G is defined as the x-detour number of G; denoted by dx(G): An x-detour set of cardinality dx(G) is called a dx-set of G: A connected x-detour set of G is an x-detour set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected x-detour set of G is the connected x-detour number of G and is denoted by cdx(G). A connected x-detour set of cardinality cdx(G) is called a cdx-set of G. A connected x-detour set Sx is called a minimal connected x-detour set if no proper subset of Sx is a connected x-detour set. The upper connected x-detour number, denoted by cd+ x (G), is defined as the maximum cardinality of a minimal connected x-detour set of G: We determine bounds for cd+ x (G) and find the same for some special classes of graphs. For any three integers a; b and c with 2 ? a < b ? c, there is a connected graph G with dx(G) = a; cdx(G) = b and cd+ x (G) = c for some vertex x in G: It is shown that for positive integers R,D and n ? 3 with R < D ? 2R; there exists a connected graph G with detour radius R; detour diameter D and cd+ x (G) = n for some vertex x in G.

scholarly journals A note on the open packing number in graphs

2018 ◽  
Vol 144 (2) ◽  
pp. 221-224
Author(s):  
Mehdi Mohammadi ◽  
Mohammad Maghasedi
Keyword(s):  
Packing Number ◽  
Open Packing
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