scholarly journals Classes of Planar Graphs with Constant Edge Metric Dimension

Complexity ◽  
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.

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

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.

scholarly journals On the Edge Metric Dimension of Certain Polyphenyl Chains

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.

scholarly journals ON METRIC DIMENSION OF FUNCTIGRAPHS

2013 ◽  
Vol 05 (04) ◽  
pp. 1250060 ◽  
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.

scholarly journals On the Geodesic Identification of Vertices in Convex Plane Graphs

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Fawaz E. Alsaadi ◽  
Muhammad Salman ◽  
Masood Ur Rehman ◽  
Abdul Rauf Khan ◽  
Jinde Cao ◽  
...  
Keyword(s):  
Shortest Path ◽  
Connected Graph ◽  
Metric Dimension ◽  
Plane Graphs ◽  
Convex Plane ◽  
Strong Metric Dimension ◽  
Minimum Number

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.

Minimal Decompositions of Complete Graphs into Subgraphs with Embeddability Properties

1969 ◽  
Vol 21 ◽  
pp. 992-1000 ◽  
Author(s):  
L. W. Beineke
Keyword(s):  
Projective Plane ◽  
Complete Graph ◽  
Planar Graphs ◽  
Complete Graphs ◽  
Minimum Number ◽  
Double Torus

Although the problem of finding the minimum number of planar graphs into which the complete graph can be decomposed remains partially unsolved, the corresponding problem can be solved for certain other surfaces. For three, the torus, the double-torus, and the projective plane, a single proof will be given to provide the solutions. The same questions will also be answered for bicomplete graphs.

scholarly journals The metric dimension of strong product graphs

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.

Adjacent vertex distinguishing edge coloring of planar graphs without 3-cycles

2020 ◽  
Vol 12 (04) ◽  
pp. 2050035
Author(s):  
Danjun Huang ◽  
Xiaoxiu Zhang ◽  
Weifan Wang ◽  
Stephen Finbow
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Edge Coloring ◽  
Discrete Math ◽  
Adjacent Vertex ◽  
Chromatic Index ◽  
Proper Edge Coloring ◽  
Minimum Number

The adjacent vertex distinguishing edge coloring of a graph [Formula: see text] is a proper edge coloring of [Formula: see text] such that the color sets of any pair of adjacent vertices are distinct. The minimum number of colors required for an adjacent vertex distinguishing edge coloring of [Formula: see text] is denoted by [Formula: see text]. It is observed that [Formula: see text] when [Formula: see text] contains two adjacent vertices of degree [Formula: see text]. In this paper, we prove that if [Formula: see text] is a planar graph without 3-cycles, then [Formula: see text]. Furthermore, we characterize the adjacent vertex distinguishing chromatic index for planar graphs of [Formula: see text] and without 3-cycles. This improves a result from [D. Huang, Z. Miao and W. Wang, Adjacent vertex distinguishing indices of planar graphs without 3-cycles, Discrete Math. 338 (2015) 139–148] that established [Formula: see text] for planar graphs without 3-cycles.

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

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.

scholarly journals On Metric Dimensions of Symmetric Graphs Obtained by Rooted Product

Mathematics ◽  
2018 ◽  
Vol 6 (10) ◽  
pp. 191 ◽  
Author(s):  
Shahid Imran ◽  
Muhammad Siddiqui ◽  
Muhammad Imran ◽  
Muhammad Hussain
Keyword(s):  
Connected Graph ◽  
The Other ◽  
Metric Dimension ◽  
Resolving Set ◽  
Minimum Cardinality ◽  
Four Dimensions ◽  
Symmetric Graphs ◽  
Metric Dimensions ◽  
Cycle Path

Let G = (V, E) be a connected graph and d(x, y) be the distance between the vertices x and y in G. A set of vertices W resolves a graph G if every vertex is uniquely determined by its vector of distances to the vertices in W. A metric dimension of G is the minimum cardinality of a resolving set of G and is denoted by dim(G). In this paper, Cycle, Path, Harary graphs and their rooted product as well as their connectivity are studied and their metric dimension is calculated. It is proven that metric dimension of some graphs is unbounded while the other graphs are constant, having three or four dimensions in certain cases.

scholarly journals Edge Version of Metric Dimension and Doubly Resolving Sets of the Necklace Graph

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 243 ◽  
Author(s):  
Jia-Bao Liu ◽  
Zohaib Zahid ◽  
Ruby Nasir ◽  
Waqas Nazeer
Keyword(s):  
Connected Graph ◽  
Metric Dimension ◽  
Necklace Graph

Consider an undirected and connected graph G = ( V G , E G ) , where V G and E G represent the set of vertices and the set of edges respectively. The concept of edge version of metric dimension and doubly resolving sets is based on the distances of edges in a graph. In this paper, we find the edge version of metric dimension and doubly resolving sets for the necklace graph.

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