The metric dimension of strong product graphs

JUAN A. RODRIGUEZ-VELAZQUEZ;  ; DOROTA KUZIAK; ISMAEL G. YERO; JOSE M. SIGARRETA;  ;  ;

doi:10.37193/cjm.2015.02.15

The metric dimension of strong product graphs

Carpathian Journal of Mathematics ◽

10.37193/cjm.2015.02.15 ◽

2015 ◽

Vol 31 (2) ◽

pp. 261-268

Author(s):

JUAN A. RODRIGUEZ-VELAZQUEZ ◽

◽

DOROTA KUZIAK ◽

ISMAEL G. YERO ◽

JOSE M. SIGARRETA ◽

...

Keyword(s):

Connected Graph ◽

Metric Dimension ◽

Np Hard ◽

Strong Product ◽

Product Graphs ◽

Metric Basis

For an ordered subset S = {s1, s2, . . . sk} of vertices in a connected graph G, the metric representation of a vertex u with respect to the set S is the k-vector r(u|S) = (dG(v, s1), dG(v, s2), . . . , dG(v, sk)), where dG(x, y) represents the distance between the vertices x and y. The set S is a metric generator for G if every two different vertices of G have distinct metric representations with respect to S. A minimum metric generator is called a metric basis for G and its cardinality, dim(G), the metric dimension of G. It is well known that the problem of finding the metric dimension of a graph is NP-Hard. In this paper we obtain closed formulae and tight bounds for the metric dimension of strong product graphs.

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On the strong metric dimension of the strong products of graphs

Open Mathematics ◽

10.1515/math-2015-0007 ◽

2015 ◽

Vol 13 (1) ◽

Cited By ~ 1

Author(s):

Dorota Kuziak ◽

Ismael G. Yero ◽

Juan A. Rodríguez-Velázquez

Keyword(s):

Connected Graph ◽

Metric Dimension ◽

Np Hard ◽

Factor Graphs ◽

Resolving Set ◽

Sharp Bounds ◽

Strong Product ◽

Product Graphs ◽

Strong Metric Dimension

AbstractLet G be a connected graph. A vertex w ∈ V.G/ strongly resolves two vertices u,v ∈ V.G/ if there exists some shortest u-w path containing v or some shortest v-w path containing u. A set S of vertices is a strong resolving set for G if every pair of vertices of G is strongly resolved by some vertex of S. The smallest cardinality of a strong resolving set for G is called the strong metric dimension of G. It is well known that the problem of computing this invariant is NP-hard. In this paper we study the problem of finding exact values or sharp bounds for the strong metric dimension of strong product graphs and express these in terms of invariants of the factor graphs.

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Metric basis and metric dimension of 1-pentagonal carbon nanocone networks

Scientific Reports ◽

10.1038/s41598-020-76516-1 ◽

2020 ◽

Vol 10 (1) ◽

Author(s):

Zafar Hussain ◽

Mobeen Munir ◽

Ashfaq Ahmad ◽

Maqbool Chaudhary ◽

Junaid Alam Khan ◽

...

Keyword(s):

Combinatorial Chemistry ◽

Connected Graph ◽

Cardinal Number ◽

Metric Dimension ◽

Molecular Topology ◽

Resolving Set ◽

Computer Chemistry ◽

Carbon Nanocones ◽

Vertex Set ◽

Metric Basis

AbstractResolving set and metric basis has become an integral part in combinatorial chemistry and molecular topology. It has a lot of applications in computer, chemistry, pharmacy and mathematical disciplines. A subset S of the vertex set V of a connected graph G resolves G if all vertices of G have different representations with respect to S. A metric basis for G is a resolving set having minimum cardinal number and this cardinal number is called the metric dimension of G. In present work, we find a metric basis and also metric dimension of 1-pentagonal carbon nanocones. We conclude that only three vertices are minimal requirement for the unique identification of all vertices in this network.

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On the metric dimension and diameter of circulant graphs with three jumps

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830918500088 ◽

2018 ◽

Vol 10 (01) ◽

pp. 1850008

Author(s):

Muhammad Imran ◽

A. Q. Baig ◽

Saima Rashid ◽

Andrea Semaničová-Feňovčíková

Keyword(s):

Positive Integer ◽

Connected Graph ◽

Metric Spaces ◽

Metric Dimension ◽

Resolving Set ◽

Circulant Graphs ◽

Infinite Class ◽

Metric Properties ◽

Minimum Number ◽

Metric Basis

Let [Formula: see text] be a connected graph and [Formula: see text] be the distance between the vertices [Formula: see text] and [Formula: see text] in [Formula: see text]. The diameter of [Formula: see text] is defined as [Formula: see text] and is denoted by [Formula: see text]. A subset of vertices [Formula: see text] is called a resolving set for [Formula: see text] if for every two distinct vertices [Formula: see text], there is a vertex [Formula: see text], [Formula: see text], such that [Formula: see text]. A resolving set containing the minimum number of vertices is called a metric basis for [Formula: see text] and the number of vertices in a metric basis is its metric dimension, denoted by [Formula: see text]. Metric dimension is a generalization of affine dimension to arbitrary metric spaces (provided a resolving set exists). Let [Formula: see text] be a family of connected graphs [Formula: see text] depending on [Formula: see text] as follows: the order [Formula: see text] and [Formula: see text]. If there exists a constant [Formula: see text] such that [Formula: see text] for every [Formula: see text] then we shall say that [Formula: see text] has bounded metric dimension, otherwise [Formula: see text] has unbounded metric dimension. If all graphs in [Formula: see text] have the same metric dimension, then [Formula: see text] is called a family of graphs with constant metric dimension. In this paper, we study the metric properties of an infinite class of circulant graphs with three generators denoted by [Formula: see text] for any positive integer [Formula: see text] and when [Formula: see text]. We compute the diameter and determine the exact value of the metric dimension of these circulant graphs.

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Edge version of metric dimension for the families of grid graphs and generalized prism graphs

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830920500378 ◽

2020 ◽

Vol 12 (03) ◽

pp. 2050037

Author(s):

Ruby Nasir ◽

Zohaib Zahid ◽

Sohail Zafar

Keyword(s):

Connected Graph ◽

Metric Dimension ◽

Grid Graphs ◽

Metric Basis

The minimum edge version of metric basis is the smallest set [Formula: see text] of edges in a connected graph [Formula: see text] such that for every pair of edges [Formula: see text] [Formula: see text][Formula: see text] there exists an edge [Formula: see text] [Formula: see text][Formula: see text] for which [Formula: see text] [Formula: see text] [Formula: see text] holds. In this paper, the families of grid graphs and generalized prism graphs have been studied for edge version of metric dimension. Edge version of metric dimension is found to be constant for both families of graphs.

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Computing Metric Dimension and Metric Basis of 2D Lattice of Alpha-Boron Nanotubes

Symmetry ◽

10.3390/sym10080300 ◽

2018 ◽

Vol 10 (8) ◽

pp. 300 ◽

Cited By ~ 5

Author(s):

Zafar Hussain ◽

Mobeen Munir ◽

Maqbool Chaudhary ◽

Shin Kang

Keyword(s):

Electronic Structure ◽

Transport Properties ◽

Present Article ◽

Connected Graph ◽

Metric Dimension ◽

Lattice Structures ◽

Resolving Set ◽

Minimum Cardinality ◽

Mathematical Sciences ◽

Metric Basis

Concepts of resolving set and metric basis has enjoyed a lot of success because of multi-purpose applications both in computer and mathematical sciences. For a connected graph G(V,E) a subset W of V(G) is a resolving set for G if every two vertices of G have distinct representations with respect to W. A resolving set of minimum cardinality is called a metric basis for graph G and this minimum cardinality is known as metric dimension of G. Boron nanotubes with different lattice structures, radii and chirality’s have attracted attention due to their transport properties, electronic structure and structural stability. In the present article, we compute the metric dimension and metric basis of 2D lattices of alpha-boron nanotubes.

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Computing Edge Metric Dimension of One-Pentagonal Carbon Nanocone

Frontiers in Physics ◽

10.3389/fphy.2021.749166 ◽

2021 ◽

Vol 9 ◽

Author(s):

Sunny Kumar Sharma ◽

Hassan Raza ◽

Vijay Kumar Bhat

Keyword(s):

Combinatorial Chemistry ◽

Connected Graph ◽

Metric Dimension ◽

Molecular Topology ◽

Resolving Set ◽

Minimum Cardinality ◽

Crucial Information ◽

Metric Basis

Minimum resolving sets (edge or vertex) have become an integral part of molecular topology and combinatorial chemistry. Resolving sets for a specific network provide crucial information required for the identification of each item contained in the network, uniquely. The distance between an edge e = cz and a vertex u is defined by d(e, u) = min{d(c, u), d(z, u)}. If d(e1, u) ≠ d(e2, u), then we say that the vertex u resolves (distinguishes) two edges e1 and e2 in a connected graph G. A subset of vertices RE in G is said to be an edge resolving set for G, if for every two distinct edges e1 and e2 in G we have d(e1, u) ≠ d(e2, u) for at least one vertex u ∈ RE. An edge metric basis for G is an edge resolving set with minimum cardinality and this cardinality is called the edge metric dimension edim(G) of G. In this article, we determine the edge metric dimension of one-pentagonal carbon nanocone (1-PCNC). We also show that the edge resolving set for 1-PCNC is independent.

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The k-Metric Dimension of a Unicyclic Graph

Mathematics ◽

10.3390/math9212789 ◽

2021 ◽

Vol 9 (21) ◽

pp. 2789

Author(s):

Alejandro Estrada-Moreno

Keyword(s):

Linear Programming ◽

Programming Problem ◽

Connected Graph ◽

Linear Programming Problem ◽

Unicyclic Graph ◽

Metric Dimension ◽

Minimum Cardinality ◽

Metric Basis

Given a connected graph G=(V(G),E(G)), a set S⊆V(G) is said to be a k-metric generator for G if any pair of different vertices in V(G) is distinguished by at least k elements of S. A metric generator of minimum cardinality among all k-metric generators is called a k-metric basis and its cardinality is the k-metric dimension of G. We initially present a linear programming problem that describes the problem of finding the k-metric dimension and a k-metric basis of a graph G. Then we conducted a study on the k-metric dimension of a unicyclic graph.

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Fractional Local Metric Dimension of Comb Product Graphs

Baghdad Science Journal ◽

10.21123/bsj.2020.17.4.1288 ◽

2020 ◽

Vol 17 (4) ◽

pp. 1288

Author(s):

Siti Aisyah ◽

Mohammad Imam Utoyo ◽

Liliek Susilowati

Keyword(s):

Connected Graph ◽

Metric Dimension ◽

Connected Graphs ◽

Product Graphs

The local resolving neighborhood of a pair of vertices for and is if there is a vertex in a connected graph where the distance from to is not equal to the distance from to , or defined by . A local resolving function of is a real valued function such that for and . The local fractional metric dimension of graph denoted by , defined by In this research, the author discusses about the local fractional metric dimension of comb product are two graphs, namely graph and graph , where graph is a connected graphs and graph is a complate graph and denoted by We get

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On the local metric dimension of t-fold wheel, Pn o Km, and generalized fan

Indonesian Journal of Combinatorics ◽

10.19184/ijc.2018.2.2.4 ◽

2018 ◽

Vol 2 (2) ◽

pp. 88

Author(s):

Rokhana Ayu Solekhah ◽

Tri Atmojo Kusmayadi

Keyword(s):

Connected Graph ◽

Ordered Set ◽

Metric Dimension ◽

Minimum Cardinality ◽

Wheel Graph ◽

Metric Set ◽

Metric Basis ◽

Fan Graph

Let G be a connected graph and let u, v ∈ V(G). For an ordered set W = {w1, w2, ..., wn} of n distinct vertices in G, the representation of a vertex v of G with respect to W is the n-vector r(v∣W) = (d(v, w1), d(v, w2), ..., d(v, wn)), where d(v, wi) is the distance between v and wi for 1 ≤ i ≤ n. The set W is a local metric set of G if r(u ∣ W) ≠ r(v ∣ W) for every pair u, v of adjacent vertices of G. The local metric set of G with minimum cardinality is called a local metric basis for G and its cardinality is called a local metric dimension, denoted by lmd(G). In this paper we determine the local metric dimension of a t-fold wheel graph, Pn ⊙ Km graph, and generalized fan graph.

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Super Edge-Connectivity of Strong Product Graphs

Journal of Interconnection Networks ◽

10.1142/s0219265917500074 ◽

2017 ◽

Vol 17 (02) ◽

pp. 1750007 ◽

Cited By ~ 1

Author(s):

ZHAO WANG ◽

YAPING MAO ◽

CHENGFU YE ◽

HAIXING ZHAO

Keyword(s):

Connected Graph ◽

Edge Connectivity ◽

Minimum Cardinality ◽

Strong Product ◽

Product Graphs ◽

Product Of Graphs

The super edge-connectivity [Formula: see text] of a connected graph G is the minimum cardinality of an edge-cut F in G such that every component of G − F contains at least two vertices. Denote by [Formula: see text] the strong product of graphs G and H. For two graphs G and H, Yang proved that [Formula: see text]. In this paper, we give another proof of this result. In particular, we determine [Formula: see text] if [Formula: see text] and [Formula: see text], where [Formula: see text] denotes the minimum edge-degree of a graph G.

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