scholarly journals The Metric Dimension and Local Metric Dimension of Relative Prime Graph

CAUCHY ◽  
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.

scholarly journals Resolving Sets and Semi-Resolving Sets in Finite Projective Planes

10.37236/2582 ◽  
2012 ◽  
Vol 19 (4) ◽  
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.

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.

On the connected metric dimension of graphs and their complements

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.

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 On odd prime labeling of graphs

2020 ◽  
Vol 3 (3) ◽  
pp. 33-40
Author(s):  
Maged Zakaria Youssef ◽  
◽  
Zainab Saad Almoreed ◽  
Keyword(s):  
Necessary Conditions ◽  
Prime Graph ◽  
Prime Graphs ◽  
Vertex Set ◽  
Prime Labeling

In this paper we give a new variation of the prime labeling. We call a graph \(G\) with vertex set \(V(G)\) has an odd prime labeling if its vertices can be labeled distinctly from the set \(\big\{1, 3, 5, ...,2\big|V(G)\big| -1\big\}\) such that for every edge \(xy\) of \(E(G)\) the labels assigned to the vertices of \(x\) and \(y\) are relatively prime. A graph that admits an odd prime labeling is called an <i>odd prime graph</i>. We give some families of odd prime graphs and give some necessary conditions for a graph to be odd prime. Finally, we conjecture that every prime graph is odd prime graph.

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 Dimensi Metrik Graf Kr+mKsr, m, r, s, En

CAUCHY ◽  
2011 ◽  
Vol 1 (4) ◽  
pp. 165
Author(s):  
Hindayani Hindayani
Keyword(s):  
Metric Dimension ◽  
Combinatorial Search ◽  
Robotic Navigation ◽  
Resolving Set ◽  
Minimum Cardinality ◽  
Metric Dimensions ◽  
Partition Graph

<div class="standard"><a id="magicparlabel-29">The concept of minimum resolving set has proved to be useful and or related to a variety of fields such as Chemistry, Robotic Navigation, and Combinatorial Search and Optimization. So that, this thesis explains the metric dimension of graph Kr + mKsr, m, r, s E N. Resolving set of a graph G is a subset of F (G) that its distance representation is distinct to all vertices of graph G. Resolving set with minimum cardinality is called minimum resolving set, and cardinal states metric dimension of G and noted with dim (G). By drawing the graph, it will be found the resolving set, minimum resolving set and the metric dimension easily. After that, formulate those metric dimensions into a theorem. This research search for the metric dimension of Kr + mKs, m &gt; 2, m,r,s E N and its outcome are dim (Kr + mK1)= m+ (r-2) and dim(Kr + mKs)= m(s-1)+(r-2). This research can be continued for determining the metric dimension of another graph, by changing the operation of its graph or partition graph.</a></div>

scholarly journals Fault-Tolerant Metric Dimension of Circulant Graphs

Mathematics ◽  
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.

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

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Jia-Bao Liu ◽  
Ali Zafari ◽  
Hassan Zarei
Keyword(s):  
Ordered Set ◽  
Undirected Graph ◽  
Metric Dimension ◽  
Np Hard ◽  
Cocktail Party ◽  
Resolving Set ◽  
Strong Metric Dimension ◽  
Vertex Set

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.

The Metric Dimension of Circulant Graphs

2017 ◽  
Vol 60 (1) ◽  
pp. 206-216 ◽  
Author(s):  
Tomáš Vetrík
Keyword(s):  
Metric Dimension ◽  
Circulant Graph ◽  
Resolving Set ◽  
Circulant Graphs ◽  
Vertex Set ◽  
Infinite Set

Abstract. A subsetWof the vertex set of a graphGis called aresolving setofGif for every pair of distinct verticesu,vofG, there isw∊Wsuch that the distance ofwanduis different from the distance ofwandv. The cardinality of a smallest resolving set is called the metric dimension ofG, denoted by dim(G). The circulant graphCn(1, 2, . . . ,t) consists of the verticesv0,v1, . . . ,vn−1and the edgesvivi+j, where 0 ≤i≤n− 1, 1 ≤j≤t(), the indices are taken modulon. Grigorious, Manuel, Miller, Rajan, and Stephen proved that dim(Cn(1, 2, . . . ,t)) ≥t+ 1 for, and they presented a conjecture saying that dim(Cn(1, 2, . . . ,t)) =t+p− 1 forn= 2tk+t+p, where 3 ≤p≤t+ 1. We disprove both statements. We show that ift≥ 4 is even, there exists an infinite set of values ofnsuch that dim(Cn(1, 2, . . . ,t)) =t. We also prove that dim(Cn(1, 2, . . . ,t)) ≤t+p/2 forn= 2tk+t+p, wheretandpare even,t≥ 4, 2 ≤p≤t, andk≥ 1.

Sign in / Sign up

Export Citation Format

Share Document