On the Strong Metric Dimension of Tetrahedral Diamond Lattice

2015 ◽  
Vol 9 (2) ◽  
pp. 201-208 ◽  
Author(s):  
Paul Manuel ◽  
Bharati Rajan ◽  
Cyriac Grigorious ◽  
Sudeep Stephen
Keyword(s):  
Diamond Lattice ◽  
Metric Dimension ◽  
Strong Metric Dimension

Metric and Strong Metric Dimension in Cozero-Divisor Graphs

2021 ◽  
Vol 18 (3) ◽  
Author(s):  
R. Nikandish ◽  
M. J. Nikmehr ◽  
M. Bakhtyiari
Keyword(s):  
Metric Dimension ◽  
Strong Metric Dimension

scholarly journals Dimensi Metrik Kuat Lokal Graf Hasil Operasi Kali Kartesian

2020 ◽  
Vol 1 (2) ◽  
pp. 64
Author(s):  
Nurma Ariska Sutardji ◽  
Liliek Susilowati ◽  
Utami Dyah Purwati
Keyword(s):  
Graph Theory ◽  
Cartesian Product ◽  
Metric Dimension ◽  
Star Graph ◽  
Product Graph ◽  
Path Graph ◽  
Strong Metric Dimension ◽  
Metric Dimensions ◽  
Development Result ◽  
Corona Product

The strong local metric dimension is the development result of a strong metric dimension study, one of the study topics in graph theory. Some of graphs that have been discovered about strong local metric dimension are path graph, star graph, complete graph, cycle graphs, and the result corona product graph. In the previous study have been built about strong local metric dimensions of corona product graph. The purpose of this research is to determine the strong local metric dimension of cartesian product graph between any connected graph G and H, denoted by dimsl (G x H). In this research, local metric dimension of G x H is influenced by local strong metric dimension of graph G and local strong metric dimension of graph H. Graph G and graph H has at least two order.

Variable neighborhood search for the strong metric dimension problem

2012 ◽  
Vol 39 ◽  
pp. 51-57 ◽  
Author(s):  
Nenad Mladenović ◽  
Jozef Kratica ◽  
Vera Kovačević-Vujčić ◽  
Mirjana Čangalović
Keyword(s):  
Variable Neighborhood Search ◽  
Neighborhood Search ◽  
Metric Dimension ◽  
Strong Metric Dimension

scholarly journals The Simultaneous Strong Resolving Graph and the Simultaneous Strong Metric Dimension of Graph Families

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 125
Author(s):  
Ismael González Yero
Keyword(s):  
Vertex Cover ◽  
Common Vertex ◽  
Metric Dimension ◽  
Minimum Cardinality ◽  
New Approach ◽  
Strong Metric Dimension ◽  
Vertex Set ◽  
Graph Families ◽  
Vertex Sets ◽  
Np Hardness

We consider in this work a new approach to study the simultaneous strong metric dimension of graphs families, while introducing the simultaneous version of the strong resolving graph. In concordance, we consider here connected graphs G whose vertex sets are represented as V ( G ) , and the following terminology. Two vertices u , v ∈ V ( G ) are strongly resolved by a vertex w ∈ V ( G ) , if there is a shortest w − v path containing u or a shortest w − u containing v. A set A of vertices of the graph G is said to be a strong metric generator for G if every two vertices of G are strongly resolved by some vertex of A. The smallest possible cardinality of any strong metric generator (SSMG) for the graph G is taken as the strong metric dimension of the graph G. Given a family F of graphs defined over a common vertex set V, a set S ⊂ V is an SSMG for F , if such set S is a strong metric generator for every graph G ∈ F . The simultaneous strong metric dimension of F is the minimum cardinality of any strong metric generator for F , and is denoted by Sd s ( F ) . The notion of simultaneous strong resolving graph of a graph family F is introduced in this work, and its usefulness in the study of Sd s ( F ) is described. That is, it is proved that computing Sd s ( F ) is equivalent to computing the vertex cover number of the simultaneous strong resolving graph of F . Several consequences (computational and combinatorial) of such relationship are then deduced. Among them, we remark for instance that we have proved the NP-hardness of computing the simultaneous strong metric dimension of families of paths, which is an improvement (with respect to the increasing difficulty of the problem) on the results known from the literature.

scholarly journals On the strong metric dimension of corona product graphs and join graphs

2013 ◽  
Vol 161 (7-8) ◽  
pp. 1022-1027 ◽  
Author(s):  
Dorota Kuziak ◽  
Ismael G. Yero ◽  
Juan A. Rodríguez-Velázquez
Keyword(s):  
Metric Dimension ◽  
Product Graphs ◽  
Strong Metric Dimension ◽  
Corona Product

scholarly journals Characterization of (Molecular) Graphs with Fractional Metric Dimension as Unity

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Muhammad Javaid ◽  
Muhammad Kamran Aslam ◽  
Abdulaziz Mohammed Alanazi ◽  
Meshari M. Aljohani
Keyword(s):  
Ordered Set ◽  
Chemical Compounds ◽  
Diamond Lattice ◽  
Theoretical Development ◽  
Metric Dimension ◽  
Test Case ◽  
Molecular Graphs ◽  
Path Graph ◽  
Extremal Values

Distance-based dimensions provide the foreground for the identification of chemical compounds that are chemically and structurally different but show similarity in different reactions. The reason behind this similarity is the occurrence of a set S of atoms and their same relative distances to some ordered set T of atoms in both compounds. In this article, the aforementioned problem is considered as a test case for characterising the (molecular) graphs bearing the fractional metric dimension (FMD) as 1. For the illustration of the theoretical development, it is shown that the FMD of path graph is unity. Moreover, we evaluated the extremal values of fractional metric dimension of a tetrahedral diamond lattice.

scholarly journals Computing strong metric dimension of some special classes of graphs by genetic algorithms

2008 ◽  
Vol 18 (2) ◽  
pp. 143-151 ◽  
Author(s):  
Jozef Kratica ◽  
Vera Kovacevic-Vujcic ◽  
Mirjana Cangalovic
Keyword(s):  
Genetic Algorithm ◽  
Genetic Algorithms ◽  
Graph Coloring ◽  
Crew Scheduling ◽  
Metric Dimension ◽  
Hard Problem ◽  
Genetic Operators ◽  
Strong Metric Dimension ◽  
Np Hard Problem ◽  
Special Classes

In this paper we consider the NP-hard problem of determining the strong metric dimension of graphs. The problem is solved by a genetic algorithm that uses binary encoding and standard genetic operators adapted to the problem. This represents the first attempt to solve this problem heuristically. We report experimental results for the two special classes of ORLIB test instances: crew scheduling and graph coloring.

scholarly journals Erratum to “On the strong metric dimension of the strong products of graphs”

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Dorota Kuziak ◽  
Ismael G. Yero ◽  
Juan A. Rodríguez-Velázquez
Keyword(s):  
Original Version ◽  
Metric Dimension ◽  
Central European Journal ◽  
Central European ◽  
Strong Metric Dimension

AbstractThe original version of the article was published in Open Mathematics (formerly Central European Journal of Mathematics) 13 (2015) 64–74. Unfortunately, the original version of this article contains a mistake: in Lemma 2.17 appears that for any C1-graph G and any graph H, β (G ⊠ H) ≤ β (G)(β(H)+1), while should be β (G ⊠ H) ≤ β(H) (β(G)+1). In this erratum we correct the lemma, its proof and some of its consequences.

scholarly journals Computing Minimal Doubly Resolving Sets and the Strong Metric Dimension of the Layer Sun Graph and the Line Graph of the Layer Sun Graph

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Jia-Bao Liu ◽  
Ali Zafari
Keyword(s):  
Connected Graph ◽  
Line Graph ◽  
Metric Dimension ◽  
Resolving Set ◽  
Minimum Cardinality ◽  
Strong Metric Dimension ◽  
Vertex Set

Let G be a finite, connected graph of order of, at least, 2 with vertex set VG and edge set EG. A set S of vertices of the graph G is a doubly resolving set for G if every two distinct vertices of G are doubly resolved by some two vertices of S. The minimal doubly resolving set of vertices of graph G is a doubly resolving set with minimum cardinality and is denoted by ψG. In this paper, first, we construct a class of graphs of order 2n+Σr=1k−2nmr, denoted by LSGn,m,k, and call these graphs as the layer Sun graphs with parameters n, m, and k. Moreover, we compute minimal doubly resolving sets and the strong metric dimension of the layer Sun graph LSGn,m,k and the line graph of the layer Sun graph LSGn,m,k.

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