On graphs with same metric and upper dimension

2020 ◽  
pp. 2150015
Author(s):  
S. Pirzada ◽  
M. Aijaz
Keyword(s):  
Discrete Math ◽  
Proper Subset ◽  
Metric Dimension ◽  
Resolving Set ◽  
Empty Graph ◽  
Finite Vector ◽  
Correct Form ◽  
Totally Disconnected ◽  
Disconnected Graph

The metric representation of a vertex [Formula: see text] of a graph [Formula: see text] is a finite vector representing distances of [Formula: see text] with respect to vertices of some ordered subset [Formula: see text]. The set [Formula: see text] is called a minimal resolving set if no proper subset of [Formula: see text] gives distinct representations for all vertices of [Formula: see text]. The metric dimension of [Formula: see text] is the cardinality of the smallest (with respect to its cardinality) minimal resolving set and upper dimension is the cardinality of the largest minimal resolving set. We show the existence of graphs for which metric dimension equals upper dimension. We found an error in a result, defining the metric dimension of join of path and totally disconnected graph, of the paper by Shahida and Sunitha [On the metric dimension of join of a graph with empty graph ([Formula: see text]), Electron. Notes Discrete Math. 63 (2017) 435–445] and we give the correct form of the theorem and its proof.

scholarly journals On resolvability of a graph associated to a finite vector space

2019 ◽  
Vol 18 (02) ◽  
pp. 1950029
Author(s):  
U. Ali ◽  
S. A. Bokhary ◽  
K. Wahid ◽  
G. Abbas
Keyword(s):  
Vector Space ◽  
Domination Number ◽  
Exchange Property ◽  
Metric Dimension ◽  
Resolving Set ◽  
Matroid Intersection ◽  
Finite Vector ◽  
Finite Vector Space ◽  
Partition Dimension ◽  
Matroid Intersection Problem

In this paper, the resolving parameters such as metric dimension and partition dimension for the nonzero component graph, associated to a finite vector space, are discussed. The exact values of these parameters are determined. It is derived that the notions of metric dimension and locating-domination number coincide in the graph. Independent sets, introduced by Boutin [Determining sets, resolving set, and the exchange property, Graphs Combin. 25 (2009) 789–806], are studied in the graph. It is shown that the exchange property holds in the graph for minimal resolving sets with some exceptions. Consequently, a minimal resolving set of the graph is a basis for a matroid with the set [Formula: see text] of nonzero vectors of the vector space as the ground set. The matroid intersection problem for two matroids with [Formula: see text] as the ground set is also solved.

scholarly journals Power graphs and exchange property for resolving sets

2019 ◽  
Vol 17 (1) ◽  
pp. 1303-1309 ◽  
Author(s):  
Ghulam Abbas ◽  
Usman Ali ◽  
Mobeen Munir ◽  
Syed Ahtsham Ul Haq Bokhary ◽  
Shin Min Kang
Keyword(s):  
Present Article ◽  
Linear Algebra ◽  
Finite Groups ◽  
Robot Navigation ◽  
Simple Graph ◽  
Exchange Property ◽  
Metric Dimension ◽  
Sufficient Condition ◽  
Resolving Set ◽  
Dihedral Groups

Abstract Classical applications of resolving sets and metric dimension can be observed in robot navigation, networking and pharmacy. In the present article, a formula for computing the metric dimension of a simple graph wihtout singleton twins is given. A sufficient condition for the graph to have the exchange property for resolving sets is found. Consequently, every minimal resolving set in the graph forms a basis for a matriod in the context of independence defined by Boutin [Determining sets, resolving set and the exchange property, Graphs Combin., 2009, 25, 789-806]. Also, a new way to define a matroid on finite ground is deduced. It is proved that the matroid is strongly base orderable and hence satisfies the conjecture of White [An unique exchange property for bases, Linear Algebra Appl., 1980, 31, 81-91]. As an application, it is shown that the power graphs of some finite groups can define a matroid. Moreover, we also compute the metric dimension of the power graphs of dihedral groups.

scholarly journals Metric Dimension Parameterized By Treewidth

Algorithmica ◽  
2021 ◽  
Author(s):  
Édouard Bonnet ◽  
Nidhi Purohit
Keyword(s):  
Computable Function ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Minimum Size ◽  
Metric Dimension ◽  
Resolving Set ◽  
Input Graph ◽  
Outerplanar Graphs ◽  
Bounded Treewidth ◽  
Fpt Algorithm

AbstractA resolving set S of a graph G is a subset of its vertices such that no two vertices of G have the same distance vector to S. The Metric Dimension problem asks for a resolving set of minimum size, and in its decision form, a resolving set of size at most some specified integer. This problem is NP-complete, and remains so in very restricted classes of graphs. It is also W[2]-complete with respect to the size of the solution. Metric Dimension has proven elusive on graphs of bounded treewidth. On the algorithmic side, a polynomial time algorithm is known for trees, and even for outerplanar graphs, but the general case of treewidth at most two is open. On the complexity side, no parameterized hardness is known. This has led several papers on the topic to ask for the parameterized complexity of Metric Dimension with respect to treewidth. We provide a first answer to the question. We show that Metric Dimension parameterized by the treewidth of the input graph is W[1]-hard. More refinedly we prove that, unless the Exponential Time Hypothesis fails, there is no algorithm solving Metric Dimension in time $$f(\text {pw})n^{o(\text {pw})}$$ f ( pw ) n o ( pw ) on n-vertex graphs of constant degree, with $$\text {pw}$$ pw the pathwidth of the input graph, and f any computable function. This is in stark contrast with an FPT algorithm of Belmonte et al. (SIAM J Discrete Math 31(2):1217–1243, 2017) with respect to the combined parameter $$\text {tl}+\Delta$$ tl + Δ , where $$\text {tl}$$ tl is the tree-length and $$\Delta$$ Δ the maximum-degree of the input graph.

scholarly journals Metric dimension of metric transform and wreath product

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.

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 Computation of the Double Metric Dimension in Convex Polytopes

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Liying Pan ◽  
Muhammad Ahmad ◽  
Zohaib Zahid ◽  
Sohail Zafar
Keyword(s):  
Source Localization ◽  
Computer Network ◽  
Convex Polytopes ◽  
The Internet ◽  
Metric Dimension ◽  
Human Beings ◽  
Source Detection ◽  
Resolving Set ◽  
Detection Problem ◽  
Rumor Spreading

A source detection problem in complex networks has been studied widely. Source localization has much importance in order to model many real-world phenomena, for instance, spreading of a virus in a computer network, epidemics in human beings, and rumor spreading on the internet. A source localization problem is to identify a node in the network that gives the best description of the observed diffusion. For this purpose, we select a subset of nodes with least size such that the source can be uniquely located. This is equivalent to find the minimal doubly resolving set of a network. In this article, we have computed the double metric dimension of convex polytopes R n and Q n by describing their minimal doubly resolving sets.

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 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.

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.

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