scholarly journals Further new results on strong resolving partitions for graphs

2020 ◽  
Vol 18 (1) ◽  
pp. 237-248 ◽  
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.

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.

scholarly journals On the Strong Partition Dimension of Graphs

10.37236/3474 ◽  
2014 ◽  
Vol 21 (3) ◽  
Author(s):  
Ismael González Yero
Keyword(s):  
Connected Graph ◽  
Minimum Cardinality ◽  
Single Vertex ◽  
Mathematical Properties ◽  
Vertex Partition ◽  
Vertex Set ◽  
Partition Dimension

We present a new style of metric generator in graphs. Specifically we introduce a metric generator based on a partition of the vertex set of a graph. The sets of the partition will work as the elements which will uniquely determine the position of each single vertex of the graph. A set $W$ of vertices of a connected graph $G$ strongly resolves two different vertices $x,y\notin W$ 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 $d_G(x,W)=\min\left\{d(x,w)\;:\;w\in W\right\}$. An ordered vertex partition $\Pi=\left\{U_1,U_2,...,U_k\right\}$ 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 set of $\Pi$. A strong resolving partition of minimum cardinality is called a strong partition basis and its cardinality the strong partition dimension. In this article we introduce the concepts of strong resolving partition and strong partition dimension and we begin with the study of its mathematical properties.

On the Strong Metric Dimension of Certain Nanostructures

2017 ◽  
Vol 14 (1) ◽  
pp. 354-358 ◽  
Author(s):  
Sathish Krishnan ◽  
Bharati Rajan ◽  
Muhammad Imran
Keyword(s):  
Connected Graph ◽  
Metric Dimension ◽  
Resolving Set ◽  
Minimum Cardinality ◽  
Strong Metric Dimension ◽  
Metric Basis

Let G(V, E) be a connected graph. A vertex w strongly resolves a pair of vertices u,v in V if there exists some shortestu–w path containing V or some shortest v–w path containing u. A set w ⊂ V of vertices is called a strong resolving set for G if every pair of vertices of V\W is strongly resolved by some vertex of w . A strong resolving set of minimum cardinality is called a strong metric basis and this cardinality is called the strong metric dimension of G. The strong metric dimension problem is to find a strong metric basis in a graph. In this paper we investigate the strong metric dimension problem for certain nanostructures.

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

Symmetry ◽  
2018 ◽  
Vol 10 (8) ◽  
pp. 300 ◽  
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.

scholarly journals The Partition Dimension of a Path Graph

2021 ◽  
Vol 13 (2) ◽  
pp. 66
Author(s):  
Vivi Ramdhani ◽  
Fathur Rahmi
Keyword(s):  
Graph Theory ◽  
Connected Graph ◽  
Ordered Set ◽  
Minimum Cardinality ◽  
Path Graph ◽  
Partition Dimension

Resolving partition is part of graph theory. This article, explains about resolving partition of the path graph, with. Given a connected graph  and  is a subset of  writen . Suppose there is , then the distance between and  is denoted in the form . There is an ordered set of -partitions of, writen then  the representation of with respect tois the  The set of partitions ofis called a resolving partition if the representation of each  to  is different. The minimum cardinality of the solving-partition to  is called the partition dimension of G which is denoted by . Before getting the partition dimension of a path graph, the first step is to look for resolving partition of the graph. Some resolving partitions of path graph,  with ,  and  are obtained. Then, the partition dimension of the path graph which is the minimum cardinality of resolving partition, namely pd (Pn)=2Resolving partition is part of graph theory. This article, explains about resolving partition of the path graph, with. Given a connected graph  and  is a subset of  writen . Suppose there is , then the distance between and  is denoted in the form . There is an ordered set of -partitions of, writen then  the representation of with respect tois the  The set of partitions ofis called a resolving partition if the representation of each  to  is different. The minimum cardinality of the solving-partition to  is called the partition dimension of G which is denoted by . Before getting the partition dimension of a path graph, the first step is to look for resolving partition of the graph. Some resolving partitionsof path graph, with ,  and  are obtained. Then, the partition dimension of the path graph which is the minimum cardinality of resolving partition, namely.

scholarly journals Resolvability in Subdivision of Circulant Networks Cn1,k

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.

On fault-tolerant partition dimension of graphs

2021 ◽  
Vol 40 (1) ◽  
pp. 1129-1135
Author(s):  
Kamran Azhar ◽  
Sohail Zafar ◽  
Agha Kashif ◽  
Zohaib Zahid
Keyword(s):  
Connected Graph ◽  
Intelligent Systems ◽  
Fault Tolerant ◽  
Robot Navigation ◽  
Natural Extension ◽  
Resolving Set ◽  
Minimum Cardinality ◽  
Sensor Networking ◽  
Partition Dimension

Fault-tolerant resolving partition is natural extension of resolving partitions which have many applications in different areas of computer sciences for example sensor networking, intelligent systems, optimization and robot navigation. For a nontrivial connected graph G (V (G) , E (G)), the partition representation of vertex v with respect to an ordered partition Π = {Si : 1 ≤ i ≤ k} of V (G) is the k-vector r ( v | Π ) = ( d ( v , S i ) ) i = 1 k , where, d (v, Si) = min {d (v, x) |x ∈ Si}, for i ∈ {1, 2, …, k}. A partition Π is said to be fault-tolerant partition resolving set of G if r (u|Π) and r (v|Π) differ by at least two places for all u ≠ v ∈ V (G). A fault-tolerant partition resolving set of minimum cardinality is called the fault-tolerant partition basis of G and its cardinality the fault-tolerant partition dimension of G denoted by P ( G ) . In this article, we will compute fault-tolerant partition dimension of families of tadpole and necklace graphs.

scholarly journals Computing Edge Metric Dimension of One-Pentagonal Carbon Nanocone

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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