Characterization of n-Vertex Graphs of Metric Dimension n − 3 by Metric Matrix

Let G = ( V ( G ) , E ( G ) ) be a connected graph. An ordered set W ⊂ V ( G ) is a resolving set for G if every vertex of G is uniquely determined by its vector of distances to the vertices in W. The metric dimension of G is the minimum cardinality of a resolving set. In this paper, we characterize the graphs of metric dimension n − 3 by constructing a special distance matrix, called metric matrix. The metric matrix makes it so a class of graph and its twin graph are bijective and the class of graph is obtained from its twin graph, so it provides a basis for the extension of graphs with respect to metric dimension. Further, the metric matrix gives a new idea of the characterization of extremal graphs based on metric dimension.

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Non-Isolated Resolving Number of Graph with Pendant Edges

Journal of Interconnection Networks ◽

10.1142/s0219265919500038 ◽

2019 ◽

Vol 19 (02) ◽

pp. 1950003

Author(s):

RIDHO ALFARISI ◽

DAFIK ◽

ARIKA INDAH KRISTIANA ◽

IKA HESTI AGUSTIN

Keyword(s):

Connected Graph ◽

Ordered Set ◽

Metric Dimension ◽

Resolving Set ◽

Minimum Cardinality ◽

Isolated Vertex

We consider V, E are respectively vertex and edge sets of a simple, nontrivial and connected graph G. For an ordered set W = {w1, w2, w3, …, wk} of vertices and a vertex v ∈ G, the ordered r(v|W) = (d(v, w1), d(v, w2), …, d(v, wk)) of k-vector is representations of v with respect to W, where d(v, w) is the distance between the vertices v and w. The set W is called a resolving set for G if distinct vertices of G have distinct representations with respect to W. The metric dimension, denoted by dim(G) is min of |W|. Furthermore, the resolving set W of graph G is called non-isolated resolving set if there is no ∀v ∈ W induced by non-isolated vertex. While a non-isolated resolving number, denoted by nr(G), is the minimum cardinality of non-isolated resolving set in graph. In this paper, we study the non isolated resolving number of graph with any pendant edges.

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On The Local Metric Dimension of Line Graph of Special Graph

CAUCHY ◽

10.18860/ca.v4i3.3694 ◽

2016 ◽

Vol 4 (3) ◽

pp. 125

Author(s):

Marsidi Marsidi ◽

Dafik Dafik ◽

Ika Hesti Agustin ◽

Ridho Alfarisi

Keyword(s):

Connected Graph ◽

Ordered Set ◽

Line Graph ◽

Metric Dimension ◽

Resolving Set ◽

Minimum Cardinality

Let G be a simple, nontrivial, and connected graph. is a representation of an ordered set of k distinct vertices in a nontrivial connected graph G. The metric code of a vertex v, where , the ordered of k-vector is representations of v with respect to W, where is the distance between the vertices v and wi for 1≤ i ≤k. Furthermore, the set W is called a local resolving set of G if for every pair u,v of adjacent vertices of G. The local metric dimension ldim(G) is minimum cardinality of W. The local metric dimension exists for every nontrivial connected graph G. In this paper, we study the local metric dimension of line graph of special graphs , namely path, cycle, generalized star, and wheel. The line graph L(G) of a graph G has a vertex for each edge of G, and two vertices in L(G) are adjacent if and only if the corresponding edges in G have a vertex in common.

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On the joint product graphs with respect to dominant metric dimension

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830921500105 ◽

2020 ◽

pp. 2150010

Author(s):

Liliek Susilowati ◽

Imroatus Sa’adah ◽

Utami Dyah Purwati

Keyword(s):

Graph Theory ◽

Connected Graph ◽

Ordered Set ◽

Dominating Set ◽

Metric Dimension ◽

Resolving Set ◽

Minimum Cardinality ◽

Joint Product ◽

Product Graphs

Some concepts in graph theory are resolving set, dominating set, and dominant metric dimension. A resolving set of a connected graph [Formula: see text] is the ordered set [Formula: see text] such that every pair of two vertices [Formula: see text] has the different representation with respect to [Formula: see text]. A Dominating set of [Formula: see text] is the subset [Formula: see text] such that for every vertex [Formula: see text] in [Formula: see text] is adjacent to at least one vertex in [Formula: see text]. A dominant resolving set of [Formula: see text] is an ordered set [Formula: see text] such that [Formula: see text] is a resolving set and a dominating set of [Formula: see text]. The minimum cardinality of a dominant resolving set is called a dominant metric dimension of [Formula: see text], denoted by [Formula: see text]. In this paper, we determine the dominant metric dimension of the joint product graphs.

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On Metric Dimensions of Symmetric Graphs Obtained by Rooted Product

Mathematics ◽

10.3390/math6100191 ◽

2018 ◽

Vol 6 (10) ◽

pp. 191 ◽

Cited By ~ 4

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.

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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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Resolvability in Subdivision of Circulant Networks Cn1,k

Discrete Dynamics in Nature and Society ◽

10.1155/2020/4197678 ◽

2020 ◽

Vol 2020 ◽

pp. 1-11

Author(s):

Jianxin Wei ◽

Syed Ahtsham Ul Haq Bokhary ◽

Ghulam Abbas ◽

Muhammad Imran

Keyword(s):

Facility Location ◽

Connected Graph ◽

Location Problems ◽

Metric Dimension ◽

Circulant Graph ◽

Resolving Set ◽

Minimum Cardinality ◽

Facility Location Problems ◽

Wide Range ◽

Circulant Networks

Circulant networks form a very important and widely explored class of graphs due to their interesting and wide-range applications in networking, facility location problems, and their symmetric properties. A resolving set is a subset of vertices of a connected graph such that each vertex of the graph is determined uniquely by its distances to that set. A resolving set of the graph that has the minimum cardinality is called the basis of the graph, and the number of elements in the basis is called the metric dimension of the graph. In this paper, the metric dimension is computed for the graph Gn1,k constructed from the circulant graph Cn1,k by subdividing its edges. We have shown that, for k=2, Gn1,k has an unbounded metric dimension, and for k=3 and 4, Gn1,k has a bounded metric dimension.

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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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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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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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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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