A central local metric dimension on acyclic and grid graph

Abstract

<abstract><p>The local metric dimension is one of many topics in graph theory with several applications. One of its applications is a new model for assigning codes to customers in delivery services. Let $ G $ be a connected graph and $ V(G) $ be a vertex set of $ G $. For an ordered set $ W = \{ x_1, x_2, \ldots, x_k\} \subseteq V(G) $, the representation of a vertex $ x $ with respect to $ W $ is $ r_G(x|W) = \{(d(x, x_1), d(x, x_2), \ldots, d(x, x_k) \} $. The set $ W $ is said to be a local metric set of $ G $ if $ r(x|W)\neq r(y|W) $ for every pair of adjacent vertices $ x $ and $ y $ in $ G $. The eccentricity of a vertex $ x $ is the maximum distance between $ x $ and all other vertices in $ G $. Among all vertices in $ G $, the smallest eccentricity is called the radius of $ G $ and a vertex whose eccentricity equals the radius is called a central vertex of $ G $. In this paper, we developed a new concept, so-called the central local metric dimension by combining the concept of local metric dimension with the central vertex of a graph. The set $ W $ is a central local metric set if $ W $ is a local metric set and contains all central vertices of $ G $. The minimum cardinality of a central local metric set is called a central local metric dimension of $ G $. In the main result, we introduce the definition of the central local metric dimension of a graph and some properties, then construct the central local metric dimensions for trees and establish results for the grid graph.</p></abstract>

A Central Local Metric Dimension of Generalized Fan Graph, Generalized Broken Fan Graph, and Cm ⊙ K¯m

“…If 𝑆 = 𝑉(𝐺) then 𝑆 is a central local metric set of 𝐺 -𝑙𝑚𝑑 𝑠 (𝐺) = 𝑛 if and only if 𝑑𝑖𝑎𝑚(𝐺) = 𝑟𝑎𝑑(𝐺)Listiana, et al in[12] also find another results of the central local metric dimension of tree 𝑇 and grid graph 𝑃 𝑛 × 𝑃 𝑚 . Let 𝑇 be a tree with 𝑑𝑖𝑎𝑚(𝑇) = 𝑘, then 𝑙𝑚𝑑 𝑠 (𝑇) = 1, for 𝑘 even, and 𝑙𝑚𝑑 𝑠 (𝑇) = 2, for 𝑘 add, where 𝑑𝑖𝑎𝑚(𝑇) is the largest eccentricity value in 𝑇 or we called it by diameter of 𝑇.…”

“…The recent research about the local metric dimension can be found in [3][4][5]. The development of local metric dimension was explored by some author, such are the local fractional metric dimension [6], the local fractional strong metric dimension [7], the local complement metric dimension [8], the local strong metric dimension [9], the dominant local metric dimension [10,11], and central local metric dimension [12].…”

“…If radius of 𝐺, denoted by 𝑟𝑎𝑑(𝐺) is a smallest eccentricity of 𝐺, then the central vertex of 𝐺 is a vertex 𝑥 with 𝑒(𝑥) = 𝑟𝑎𝑑(𝐺). A set 𝑆 ⊆ 𝑉(𝐺) is called a central set of 𝐺 if the element of 𝑆 is all central vertices in 𝐺 [12]. Further, let 𝑊 ⊆ 𝑉(𝐺) be a local metric set of 𝐺, if 𝑆 ⊆ 𝑊, then 𝑊 is called a central local metric set of 𝐺.…”

“…Further, let 𝑊 ⊆ 𝑉(𝐺) be a local metric set of 𝐺, if 𝑆 ⊆ 𝑊, then 𝑊 is called a central local metric set of 𝐺. A central local metric set with minimal cardinality is called a central local basis set of 𝐺 and its cardinality is called central local metric dimension of 𝐺, denoted by 𝑙𝑚𝑑 𝑠 (𝐺) [12]. Theorem 1 explained the lower and upper bound of the central local metric set of 𝐺 with some characteristic of central local metric dimension of 𝐺. Theorem 1.…”

“…Theorem 1 explained the lower and upper bound of the central local metric set of 𝐺 with some characteristic of central local metric dimension of 𝐺. Theorem 1. [12] Let 𝐺 be a connected graph with ordo 𝑛 and 𝑆 is a central set of ̅̅̅̅ . Some theorem and lemma that support our result explained in method section.…”