Metric Dimension, Minimal Doubly Resolving Sets, and the Strong Metric Dimension for Jellyfish Graph and Cocktail Party Graph

Let Γ be a simple connected undirected graph with vertex set VΓ and edge set EΓ. The metric dimension of a graph Γ is the least number of vertices in a set with the property that the list of distances from any vertex to those in the set uniquely identifies that vertex. For an ordered subset W=w1,w2,…,wk of vertices in a graph Γ and a vertex v of Γ, the metric representation of v with respect to W is the k-vector rvW=dv,w1,dv,w2,…,dv,wk. If every pair of distinct vertices of Γ have different metric representations, then the ordered set W is called a resolving set of Γ. It is known that the problem of computing this invariant is NP-hard. In this paper, we consider the problem of determining the cardinality ψΓ of minimal doubly resolving sets of Γ and the strong metric dimension for the jellyfish graph JFGn,m and the cocktail party graph CPk+1.

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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 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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On enhanced power graphs of certain groups

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830920500998 ◽

2020 ◽

pp. 2050099

Author(s):

Sandeep Dalal ◽

Jitender Kumar

Keyword(s):

Cyclic Subgroup ◽

Undirected Graph ◽

Minimum Degree ◽

Independence Number ◽

Metric Dimension ◽

Matching Number ◽

Graph Invariants ◽

Power Graph ◽

Strong Metric Dimension ◽

Vertex Set

The enhanced power graph [Formula: see text] of a group [Formula: see text] is a simple undirected graph with vertex set [Formula: see text] and two distinct vertices [Formula: see text] are adjacent if both [Formula: see text] and [Formula: see text] belongs to same cyclic subgroup of [Formula: see text]. In this paper, we obtain various graph invariants viz. independence number, minimum degree and matching number of [Formula: see text], where [Formula: see text] is the dicyclic group or a class of groups of order [Formula: see text]. If [Formula: see text] is any of these groups, we prove that [Formula: see text] is perfect and then obtain its strong metric dimension.

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Metric dimension of metric transform and wreath product

Carpathian Mathematical Publications ◽

10.15330/cmp.11.2.418-421 ◽

2019 ◽

Vol 11 (2) ◽

pp. 418-421

Author(s):

B.S. Ponomarchuk

Keyword(s):

Wreath Product ◽

Metric Space ◽

Metric Spaces ◽

Metric Dimension ◽

Hard Problem ◽

Np Hard ◽

Resolving Set ◽

Np Hard Problem ◽

Metric Transform ◽

Empty Subset

Let $(X,d)$ be a metric space. A non-empty subset $A$ of the set $X$ is called resolving set of the metric space $(X,d)$ if for two arbitrary not equal points $u,v$ from $X$ there exists an element $a$ from $A$, such that $d(u,a) \neq d(v,a)$. The smallest of cardinalities of resolving subsets of the set $X$ is called the metric dimension $md(X)$ of the metric space $(X,d)$. In general, finding the metric dimension is an NP-hard problem. In this paper, metric dimension for metric transform and wreath product of metric spaces are provided. It is shown that the metric dimension of an arbitrary metric space is equal to the metric dimension of its metric transform.

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Resolving Sets and Semi-Resolving Sets in Finite Projective Planes

The Electronic Journal of Combinatorics ◽

10.37236/2582 ◽

2012 ◽

Vol 19 (4) ◽

Cited By ~ 8

Author(s):

Tamás Héger ◽

Marcella Takáts

Keyword(s):

Projective Plane ◽

Finite Projective Plane ◽

Projective Planes ◽

The Other ◽

Blocking Set ◽

Metric Dimension ◽

Desarguesian Projective Plane ◽

Resolving Set ◽

Incidence Graph ◽

Vertex Set

In a graph $\Gamma=(V,E)$ a vertex $v$ is resolved by a vertex-set $S=\{v_1,\ldots,v_n\}$ if its (ordered) distance list with respect to $S$, $(d(v,v_1),\ldots,d(v,v_n))$, is unique. A set $A\subset V$ is resolved by $S$ if all its elements are resolved by $S$. $S$ is a resolving set in $\Gamma$ if it resolves $V$. The metric dimension of $\Gamma$ is the size of the smallest resolving set in it. In a bipartite graph a semi-resolving set is a set of vertices in one of the vertex classes that resolves the other class.We show that the metric dimension of the incidence graph of a finite projective plane of order $q\geq 23$ is $4q-4$, and describe all resolving sets of that size. Let $\tau_2$ denote the size of the smallest double blocking set in PG$(2,q)$, the Desarguesian projective plane of order $q$. We prove that for a semi-resolving set $S$ in the incidence graph of PG$(2,q)$, $|S|\geq \min \{2q+q/4-3, \tau_2-2\}$ holds. In particular, if $q\geq9$ is a square, then the smallest semi-resolving set in PG$(2,q)$ has size $2q+2\sqrt{q}$. As a corollary, we get that a blocking semioval in PG$(2, q)$, $q\geq 4$, has at least $9q/4-3$ points. A corrigendum was added to this paper on March 3, 2017.

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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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The Simultaneous Strong Resolving Graph and the Simultaneous Strong Metric Dimension of Graph Families

Mathematics ◽

10.3390/math8010125 ◽

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.

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On the connected metric dimension of graphs and their complements

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830921500592 ◽

2020 ◽

pp. 2150059

Author(s):

Eunjeong Yi

Keyword(s):

Unicyclic Graph ◽

Metric Dimension ◽

Resolving Set ◽

Dimension Formula ◽

Vertex Set

Let [Formula: see text] be a graph with vertex set [Formula: see text], and let [Formula: see text] denote the length of a shortest [Formula: see text] path in [Formula: see text]. A set [Formula: see text] is called a connected resolving set of [Formula: see text] if, for any distinct [Formula: see text], there exists a vertex [Formula: see text] such that [Formula: see text], and the subgraph of [Formula: see text] induced by [Formula: see text] is connected. The connected metric dimension, [Formula: see text], of [Formula: see text] is the minimum of the cardinalities over all connected resolving sets of [Formula: see text]. For a graph [Formula: see text] and its complement [Formula: see text], each of order [Formula: see text] and connected, we conjecture that [Formula: see text]; if [Formula: see text] is a tree or a unicyclic graph, we prove the conjecture and characterize graphs achieving equality.

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The Metric Dimension and Local Metric Dimension of Relative Prime Graph

CAUCHY ◽

10.18860/ca.v6i3.10103 ◽

2020 ◽

Vol 6 (3) ◽

pp. 149-161

Author(s):

Inna Kuswandari ◽

Fatmawati Fatmawati ◽

Mohammad Imam Utoyo

Keyword(s):

Prime Graph ◽

Metric Dimension ◽

Resolving Set ◽

Prime Graphs ◽

Vertex Set ◽

Integer Rings ◽

Metric Dimensions ◽

Relative Prime

This study aims to determine the value of metric dimensions and local metric dimensions of relative prime graphs formed from modulo integer rings, namely . As a vertex set is and if and are relatively prime. By finding the pattern elements of resolving set and local resolving set, it can be shown the value of the metric dimension and the local metric dimension of graphs are and respectively, where is the number of vertices groups that formed multiple 2,3, … , and is the cardinality of set . This research can be developed by determining the value of the fractional metric dimension, local fractional metric dimension and studying the advanced properties of graphs related to their forming rings.Key Words : metric dimension; modulo ; relative prime graph; resolving set; rings.

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Fault-Tolerant Metric Dimension of Circulant Graphs

Mathematics ◽

10.3390/math10010124 ◽

2022 ◽

Vol 10 (1) ◽

pp. 124

Author(s):

Laxman Saha ◽

Rupen Lama ◽

Kalishankar Tiwary ◽

Kinkar Chandra Das ◽

Yilun Shang

Keyword(s):

Connected Graph ◽

Fault Tolerant ◽

Metric Dimension ◽

Resolving Set ◽

Circulant Graphs ◽

Minimum Cardinality ◽

Vertex Set

Let G be a connected graph with vertex set V(G) and d(u,v) be the distance between the vertices u and v. A set of vertices S={s1,s2,…,sk}⊂V(G) is called a resolving set for G if, for any two distinct vertices u,v∈V(G), there is a vertex si∈S such that d(u,si)≠d(v,si). A resolving set S for G is fault-tolerant if S\{x} is also a resolving set, for each x in S, and the fault-tolerant metric dimension of G, denoted by β′(G), is the minimum cardinality of such a set. The paper of Basak et al. on fault-tolerant metric dimension of circulant graphs Cn(1,2,3) has determined the exact value of β′(Cn(1,2,3)). In this article, we extend the results of Basak et al. to the graph Cn(1,2,3,4) and obtain the exact value of β′(Cn(1,2,3,4)) for all n≥22.

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