Determining the edge metric dimension of the generalized Petersen graph P(n, 3)

2021 ◽  
Author(s):  
David G. L. Wang ◽  
Monica M. Y. Wang ◽  
Shiqiang Zhang
Keyword(s):  
Petersen Graph ◽  
Metric Dimension ◽  
Generalized Petersen Graph

scholarly journals The K -Size Edge Metric Dimension of Graphs

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Tanveer Iqbal ◽  
Muhammad Naeem Azhar ◽  
Syed Ahtsham Ul Haq Bokhary
Keyword(s):  
Bipartite Graph ◽  
Connected Graph ◽  
Complete Bipartite Graph ◽  
Petersen Graph ◽  
Metric Dimension ◽  
Resolving Set ◽  
Generalized Petersen Graph

In this paper, a new concept k -size edge resolving set for a connected graph G in the context of resolvability of graphs is defined. Some properties and realizable results on k -size edge resolvability of graphs are studied. The existence of this new parameter in different graphs is investigated, and the k -size edge metric dimension of path, cycle, and complete bipartite graph is computed. It is shown that these families have unbounded k -size edge metric dimension. Furthermore, the k-size edge metric dimension of the graphs Pm □ Pn, Pm □ Cn for m, n ≥ 3 and the generalized Petersen graph is determined. It is shown that these families of graphs have constant k -size edge metric dimension.

scholarly journals On the Local Adjacency Metric Dimension of Generalized Petersen Graphs

CAUCHY ◽  
2019 ◽  
Vol 6 (1) ◽  
pp. 10
Author(s):  
Marsidi Marsidi ◽  
Dafik Dafik ◽  
Ika Hesti Agustin ◽  
Ridho Alfarisi
Keyword(s):  
Petersen Graph ◽  
Metric Dimension ◽  
Resolving Set ◽  
Generalized Petersen Graph ◽  
Generalized Petersen Graphs ◽  
Neighboring Vertex ◽  
Petersen Graphs ◽  
Metric Basis ◽  
Order Set

<p class="Abstract">The local adjacency metric dimension is one of graph topic. Suppose there are three neighboring vertex , ,  in path . Path  is called local if  where each has representation: a is not equals  and  may equals to . Let’s say, .  For an order set of vertices , the adjacency representation of  with respect to  is the ordered -tuple , where  represents the adjacency distance . The distance  defined by 0 if , 1 if  adjacent with , and 2 if  does not adjacent with . The set  is a local adjacency resolving set of  if for every two distinct vertices ,  and  adjacent with y then . A minimum local adjacency resolving set in  is called local adjacency metric basis. The cardinality of vertices in the basis is a local adjacency metric dimension of , denoted by . Next, we investigate the local adjacency metric dimension of generalized petersen graph.</p>

The Generalized Connectivity of Generalized Petersen Graph

2021 ◽  
Author(s):  
Yuan Si ◽  
Ping Li ◽  
Yuzhi Xiao ◽  
Jinxia Liang
Keyword(s):  
Steiner Trees ◽  
Petersen Graph ◽  
Generalized Petersen Graph ◽  
Edge Disjoint ◽  
Generalized Connectivity ◽  
Vertex Set

For a vertex set [Formula: see text] of [Formula: see text], we use [Formula: see text] to denote the maximum number of edge-disjoint Steiner trees of [Formula: see text] such that any two of such trees intersect in [Formula: see text]. The generalized [Formula: see text]-connectivity of [Formula: see text] is defined as [Formula: see text]. We get that for any generalized Petersen graph [Formula: see text] with [Formula: see text], [Formula: see text] when [Formula: see text]. We give the values of [Formula: see text] for Petersen graph [Formula: see text], where [Formula: see text], and the values of [Formula: see text] for generalized Petersen graph [Formula: see text], where [Formula: see text] and [Formula: see text].

Total irregularity strength of disjoint union of isomorphic copies of generalized Petersen graph

2017 ◽  
Vol 09 (06) ◽  
pp. 1750071
Author(s):  
Muhammad Naeem ◽  
Muhammad Kamran Siddiqui
Keyword(s):  
Disjoint Union ◽  
Petersen Graph ◽  
Generalized Petersen Graph ◽  
Irregularity Strength

Let [Formula: see text] be a graph. A total labeling [Formula: see text] is called totally irregular total [Formula: see text]-labeling of [Formula: see text] if every two distinct vertices [Formula: see text] and [Formula: see text] in [Formula: see text] satisfy [Formula: see text] and every two distinct edges [Formula: see text] and [Formula: see text] in [Formula: see text] satisfy [Formula: see text] where [Formula: see text] and [Formula: see text] The minimum [Formula: see text] for which a graph [Formula: see text] has a totally irregular total [Formula: see text]-labeling is called the total irregularity strength of [Formula: see text], denoted by [Formula: see text] In this paper, we determined the total irregularity strength of disjoint union of [Formula: see text] isomorphic copies of generalized Petersen graph.

scholarly journals Eternal Domination of Generalized Petersen Graph

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Ramy Shaheen ◽  
Ali Kassem
Keyword(s):  
Infinite Series ◽  
Dominating Set ◽  
Domination Number ◽  
Upper Bounds ◽  
Adjacent Vertex ◽  
Petersen Graph ◽  
Generalized Petersen Graph ◽  
Dominating Set Problem

An eternal dominating set of a graph G is a set of guards distributed on the vertices of a dominating set so that each vertex can be occupied by one guard only. These guards can defend any infinite series of attacks; an attack is defended by moving one guard along an edge from its position to the attacked vertex. We consider the “all guards move” of the eternal dominating set problem, in which one guard has to move to the attacked vertex and all the remaining guards are allowed to move to an adjacent vertex or stay in their current positions after each attack in order to form a dominating set on the graph and at each step can be moved after each attack. The “all guards move model” is called the m -eternal domination model. The size of the smallest m -eternal dominating set is called the m -eternal domination number and is denoted by γ m ∞ G . In this paper, we find γ m ∞ P n , 1 and γ m ∞ P n , 3 for n ≡ 0   mod   4 . We also find upper bounds for γ m ∞ P n , 2 and γ m ∞ P n , 3 when n is arbitrary.

Perfect 2-colorings of the generalized Petersen graph

2016 ◽  
Vol 126 (3) ◽  
pp. 289-294 ◽  
Author(s):  
MEHDI ALAEIYAN ◽  
HAMED KARAMI
Keyword(s):  
Petersen Graph ◽  
Generalized Petersen Graph
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