On the Geodesic Identification of Vertices in Convex Plane Graphs

A shortest path between two vertices u and v in a connected graph G is a u − v geodesic. A vertex w of G performs the geodesic identification for the vertices in a pair u , v if either v belongs to a u − w geodesic or u belongs to a v − w geodesic. The minimum number of vertices performing the geodesic identification for each pair of vertices in G is called the strong metric dimension of G . In this paper, we solve the strong metric dimension problem for three convex plane graphs by performing the geodesic identification of their vertices.

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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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On the Edge Metric Dimension of Certain Polyphenyl Chains

Journal of Chemistry ◽

10.1155/2021/3855172 ◽

2021 ◽

Vol 2021 ◽

pp. 1-6

Author(s):

Muhammad Ahsan ◽

Zohaib Zahid ◽

Dalal Alrowaili ◽

Aiyared Iampan ◽

Imran Siddique ◽

...

Keyword(s):

Graph Theory ◽

Connected Graph ◽

Metric Dimension ◽

Chain Graph ◽

Minimum Number ◽

Valence Bonds

The most productive application of graph theory in chemistry is the representation of molecules by the graphs, where vertices and edges of graphs are the atoms and valence bonds between a pair of atoms, respectively. For a vertex w and an edge f = c 1 c 2 of a connected graph G , the minimum number from distances of w with c 1 and c 2 is called the distance between w and f . If for every two distinct edges f 1 , f 2 ∈ E G , there always exists w 1 ∈ W E ⊆ V G such that d f 1 , w 1 ≠ d f 2 , w 1 , then W E is named as an edge metric generator. The minimum number of vertices in W E is known as the edge metric dimension of G . In this paper, we calculate the edge metric dimension of ortho-polyphenyl chain graph O n , meta-polyphenyl chain graph M n , and the linear [n]-tetracene graph T n and also find the edge metric dimension of para-polyphenyl chain graph L n . It has been proved that the edge metric dimension of O n , M n , and T n is bounded, while L n is unbounded.

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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 ◽

10.1155/2020/6267072 ◽

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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ON METRIC DIMENSION OF FUNCTIGRAPHS

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830912500607 ◽

2013 ◽

Vol 05 (04) ◽

pp. 1250060 ◽

Cited By ~ 2

Author(s):

LINDA EROH ◽

CONG X. KANG ◽

EUNJEONG YI

Keyword(s):

Connected Graph ◽

Metric Dimension ◽

Complete Graphs ◽

Minimum Number ◽

Vertex Set

The metric dimension of a graph G, denoted by dim (G), is the minimum number of vertices such that each vertex is uniquely determined by its distances to the chosen vertices. Let G1and G2be disjoint copies of a graph G and let f : V(G1) → V(G2) be a function. Then a functigraphC(G, f) = (V, E) has the vertex set V = V(G1) ∪ V(G2) and the edge set E = E(G1) ∪ E(G2) ∪ {uv | v = f(u)}. We study how metric dimension behaves in passing from G to C(G, f) by first showing that 2 ≤ dim (C(G, f)) ≤ 2n - 3, if G is a connected graph of order n ≥ 3 and f is any function. We further investigate the metric dimension of functigraphs on complete graphs and on cycles.

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BI-DIMENSI METRIK DARI GRAF ANTIPRISMA

Majalah Ilmiah Matematika dan Statistika ◽

10.19184/mims.v20i2.19639 ◽

2020 ◽

Vol 20 (2) ◽

pp. 53

Author(s):

Hendy Hendy ◽

M. Ismail Marzuki

Keyword(s):

Shortest Path ◽

Connected Graph ◽

Metric Dimension ◽

Resolving Set ◽

Minimum Cardinality

Let G = (V, E) be a simple and connected graph. For each x ∈ V(G), it is associated with a vector pair (a, b), denoted by S x , corresponding to subset S = {s1 , s2 , ... , s k } ⊆ V(G), with a = (d(x, s1 ), d(x, s2 ), ... , d(x, s k )) and b = (δ(x, s1 ), δ(x, s2 ), ... , δ(x, s k )). d(v, s) is the length of shortest path from vertex v to s, and δ(v, s) is the length of the furthest path from vertex v to s. The set S is called the bi-resolving set in G if S x ≠ S y for any two distinct vertices x, y ∈ V(G). The bi- metric dimension of graph G, denoted by β b (G), is the minimum cardinality of the bi-resolving set in graph G. In this study we analyze bi-metric dimension in the antiprism graph (A n ). From the analysis that has been done, it is obtained the result that bi-metric dimension of graph A n , β b (A n ) is 3. Keywords: Antiprism graph, bi-metric dimension, bi-resolving set. .

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Classes of Planar Graphs with Constant Edge Metric Dimension

Complexity ◽

10.1155/2021/5599274 ◽

2021 ◽

Vol 2021 ◽

pp. 1-10

Author(s):

Changcheng Wei ◽

Muhammad Salman ◽

Syed Shahzaib ◽

Masood Ur Rehman ◽

Juanyan Fang

Keyword(s):

Connected Graph ◽

Planar Graphs ◽

Metric Dimension ◽

Minimum Number

The number of edges in a shortest walk (without repetition of vertices) from one vertex to another vertex of a connected graph G is known as the distance between them. For a vertex x and an edge e = a b in G , the minimum number from distances of x with a and b is said to be the distance between x and e . A vertex x is said to distinguish (resolves) two distinct edges e 1 and e 2 if the distance between x and e 1 is different from the distance between x and e 2 . A set X of vertices in a connected graph G is an edge metric generator for G if every two edges of G are distinguished by some vertex in X . The number of vertices in such a smallest set X is known as the edge metric dimension of G . In this article, we solve the edge metric dimension problem for certain classes of planar graphs.

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The Strong Resolving Graph and the Strong Metric Dimension of Cactus Graphs

Mathematics ◽

10.3390/math8081266 ◽

2020 ◽

Vol 8 (8) ◽

pp. 1266

Author(s):

Dorota Kuziak

Keyword(s):

Structural Properties ◽

Connected Graph ◽

Metric Dimension ◽

Resolving Set ◽

Strong Metric Dimension ◽

Cactus Graphs ◽

The Way

A vertex w of a connected graph G strongly resolves two distinct vertices u,v∈V(G), if there is a shortest u,w path containing v, or a shortest v,w path containing u. A set S of vertices of G is a strong resolving set for G if every two distinct vertices of G are strongly resolved by a vertex of S. The smallest cardinality of a strong resolving set for G is called the strong metric dimension of G. To study the strong metric dimension of graphs, a very important role is played by a structure of graphs called the strong resolving graph In this work, we obtain the strong metric dimension of some families of cactus graphs, and along the way, we give several structural properties of the strong resolving graphs of the studied families of cactus graphs.

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Further new results on strong resolving partitions for graphs

Open Mathematics ◽

10.1515/math-2020-0142 ◽

2020 ◽

Vol 18 (1) ◽

pp. 237-248 ◽

Cited By ~ 3

Author(s):

Dorota Kuziak ◽

Ismael G. Yero

Keyword(s):

Connected Graph ◽

Metric Dimension ◽

Minimum Cardinality ◽

Vertex Partition ◽

Strong Metric Dimension ◽

Partition Dimension

Abstract A set W of vertices of a connected graph G strongly resolves two different vertices x, y ∉ W if either d G (x, W) = d G (x, y) + d G (y, W) or d G (y, W) = d G (y, x) + d G (x, W), where d G (x, W) = min{d(x,w): w ∈ W} and d(x,w) represents the length of a shortest x − w path. An ordered vertex partition Π = {U 1, U 2,…,U k } of a graph G is a strong resolving partition for G, if every two different vertices of G belonging to the same set of the partition are strongly resolved by some other set of Π. The minimum cardinality of any strong resolving partition for G is the strong partition dimension of G. In this article, we obtain several bounds and closed formulae for the strong partition dimension of some families of graphs and give some realization results relating the strong partition dimension, the strong metric dimension and the order of 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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Bounds on the sum of broadcast domination number and strong metric dimension of graphs

Discrete Mathematics Algorithms and Applications ◽

10.1142/s179383092050010x ◽

2020 ◽

Vol 12 (01) ◽

pp. 2050010

Author(s):

Eunjeong Yi

Keyword(s):

Open Problem ◽

Connected Graph ◽

Domination Number ◽

Metric Dimension ◽

Unicyclic Graphs ◽

Dimension Formula ◽

Strong Metric Dimension ◽

Vertex Set ◽

Number Formula

Let [Formula: see text] be a connected graph of order at least two with vertex set [Formula: see text]. For [Formula: see text], let [Formula: see text] denote the length of an [Formula: see text] geodesic in [Formula: see text]. A function [Formula: see text] is called a dominating broadcast function of [Formula: see text] if, for each vertex [Formula: see text], there exists a vertex [Formula: see text] such that [Formula: see text] and [Formula: see text], and the broadcast domination number, [Formula: see text], of [Formula: see text] is the minimum of [Formula: see text] over all dominating broadcast functions [Formula: see text] of [Formula: see text]. For [Formula: see text], let [Formula: see text] denote the set of vertices [Formula: see text] such that either [Formula: see text] lies on a [Formula: see text] geodesic or [Formula: see text] lies on a [Formula: see text] geodesic of [Formula: see text]. Let [Formula: see text] be a function and, for any [Formula: see text], let [Formula: see text]. We say that [Formula: see text] is a strong resolving function of [Formula: see text] if [Formula: see text] for every pair of distinct vertices [Formula: see text], and the strong metric dimension, [Formula: see text], of [Formula: see text] is the minimum of [Formula: see text] over all strong resolving functions [Formula: see text] of [Formula: see text]. For any connected graph [Formula: see text], we show that [Formula: see text]; we characterize [Formula: see text] satisfying [Formula: see text] equals two and three, respectively, and characterize unicyclic graphs achieving [Formula: see text]. For any tree [Formula: see text] of order at least three, we show that [Formula: see text], and characterize trees achieving equality. Moreover, for a tree [Formula: see text] of order [Formula: see text], we obtain the results that [Formula: see text] if [Formula: see text] is central, and that [Formula: see text] if [Formula: see text] is bicentral; in each case, we characterize trees achieving equality. We conclude this paper with some remarks and an open problem.

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