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Journal ArticleDOI

Fractional Metric Dimension of Tree and Unicyclic Graph

01 Jan 2015-Procedia Computer Science (Elsevier)-Vol. 74, pp 47-52
TL;DR: The fractional metric dimension of G, where G is a tree or G is an unicyclic graph, is determined by determining the minimal value of f (V(G)) for all resolving functions f of G.
About: This article is published in Procedia Computer Science.The article was published on 2015-01-01 and is currently open access. It has received 12 citations till now. The article focuses on the topics: Bound graph & Graph power.
Citations
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Journal ArticleDOI
TL;DR: In this article, the authors consider a comb product between two connected graphs and determine the fractional metric dimension of the comb product, where the vertex vertices are resolved by a real resolving function.
Abstract: A vertex $z$ in a connected graph $G$ \textit{resolves} two vertices $u$ and $v$ in $G$ if $d_G(u,z) eq d_G(v,z)$. \ A set of vertices $R_G\{u,v\}$ is a set of all resolving vertices of $u$ and $v$ in $G$. \ For every two distinct vertices $u$ and $v$ in $G$, a \textit{resolving function} $f$ of $G$ is a real function $f:V(G)\rightarrow[0,1]$ such that $f(R_G\{u,v\})\geq1$. \ The minimum value of $f(V(G))$ from all resolving functions $f$ of $G$ is called the \textit{fractional metric dimension} of $G$. \ In this paper, we consider a graph which is obtained by the comb product between two connected graphs $G$ and $H$, denoted by $G\rhd_o H$. \ For any connected graphs $G$, we determine the fractional metric dimension of $G\rhd_o H$ where $H$ is a connected graph having a stem or a major vertex.

19 citations


Cites background from "Fractional Metric Dimension of Tree..."

  • ...Meanwhile, the fractional metric dimension of trees and unicyclic graphs can be seen in [12]....

    [...]

Journal ArticleDOI
TL;DR: In this article, the LF-metric dimension of generalized gear networks is studied, and various results for the generalized gear network are obtained in the form of exact values and sharp bounds under certain conditions.
Abstract: The parameter of distance in the theory of networks plays a key role to study the different structural properties of the understudy networks or graphs such as symmetry, assortative, connectivity, and clustering. For the purpose, with the help of the parameter of distance, various types of metric dimensions have been defined to find the locations of machines (or robots) with respect to the minimum consumption of time, the shortest distance among the destinations, and the lesser number of utilized nodes as places of the objects. In this article, the latest derived form of metric dimension called as LF-metric dimension is studied, and various results for the generalized gear networks are obtained in the form of exact values and sharp bounds under certain conditions. The LF-metric dimension of some particular cases of generalized gear networks (called as generalized wheel networks) is also illustrated. Moreover, the bounded and unboundedness of the LF-metric dimension for all obtained results is also presented.

5 citations

Posted Content
TL;DR: In this paper, the fractional local metric dimension of graphs has been introduced and its properties and bounds have been studied, and some characterization results have been obtained for strong and cartesian product of graphs.
Abstract: The fractional versions of graph theoretic-invariants multiply the range of applications in scheduling, assignment and operational research problems. In this paper, we introduce the fractional version of local metric dimension of graphs. The local resolving neighborhood $L(xy)$ of an edge $xy$ of a graph $G$ is the set of those vertices in $G$ which resolve the vertices $x$ and $y$. A function $f:V(G)\rightarrow[0, 1]$ is a local resolving function of $G$ if $f(L(xy))\geq1$ for all edges $xy$ in $G$. The minimum value of $f(V(G))$ among all local resolving functions $f$ of $G$ is the fractional local metric dimension of $G$. We study the properties and bounds of fractional local metric dimension of graphs and give some characterization results. We determine the fractional local metric dimension of strong and cartesian product of graphs.

5 citations


Cites background from "Fractional Metric Dimension of Tree..."

  • ...The fractional metric dimension of graphs and graph products has also been studied in [1, 9, 10, 11, 14, 19]....

    [...]

TL;DR: This paper studies the latest form of metric dimension called fractional metric dimension of some connected networks such as circular diagonal ladder, double sun power, and double path networks.
Abstract: Metric dimension is an eƒective tool to study diƒerent distance-based problems in the …eld of telecommunication, robotics, computer networking, integer programming, chemistry, and electrical networking. In this paper, we study the latest form of metric dimension called fractional metric dimension of some connected networks such as circular diagonal ladder, double sun ‡ower, and double path networks.

3 citations

Journal ArticleDOI
TL;DR: In this paper , the authors studied the latest form of metric dimension called fractional metric dimension of some connected networks such as circular diagonal ladder, double sun flower, and double path networks.
Abstract: Metric dimension is an effective tool to study different distance-based problems in the field of telecommunication, robotics, computer networking, integer programming, chemistry, and electrical networking. In this paper, we study the latest form of metric dimension called fractional metric dimension of some connected networks such as circular diagonal ladder, double sun flower, and double path networks.

2 citations

References
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Journal ArticleDOI
TL;DR: Bounds on dim(G) are presented in terms of the order and the diameter of G and it is shown that dim(H)⩽dim(H×K2)⦽dim (H)+1 for every connected graph H.

821 citations

Journal ArticleDOI
TL;DR: The main new results are that the metric dimension of a graph with n nodes can be approximated in polynomial time within a factor of O(log n), and some properties of graphs with metric dimension two are presented.

727 citations

Journal ArticleDOI
TL;DR: In this paper, a combinatorial detection problem is considered in the context of the Combinatory Detection Problem (CDP) and it is shown that the problem is NP-hard.
Abstract: (1963). A Combinatory Detection Problem. The American Mathematical Monthly: Vol. 70, No. 10, pp. 1066-1070.

177 citations

Journal ArticleDOI
TL;DR: The fractional metric dimension of G is defined as dim"f(G)=min{|g|:g is a minimal resolving function of G}, where |g|=@?"v"@?"Vg(v), and this parameter is studied.

55 citations