The Simultaneous Local Metric Dimension of Graph Families

Abstract

Abstract:In a graph G = (V, E), a vertex v ∈ V is said to distinguish two vertices x and y if v, y). A set S ⊆ V is said to be a local metric generator for G if any pair of adjacent vertices of G is distinguished by some element of S. A minimum local metric generator is called a local metric basis and its cardinality the local metric dimension of G. A set S ⊆ V is said to be a simultaneous local metric generator for a graph family G = {G 1 , G 2 , . . . , G k }, defined on a common vertex set, if it is a local metric generator for every graph of the family. A minimum simultaneous local metric generator is called a simultaneous local metric basis and its cardinality the simultaneous local metric dimension of G. We study the properties of simultaneous local metric generators and bases, obtain closed formulae or tight bounds for the simultaneous local metric dimension of several graph families and analyze the complexity of computing this parameter.

Metric dimension related parameters in graphs: A survey on combinatorial, computational and applied results

“…Among them, we can find for instance the local fractional metric dimension [5], the adjacency local metric dimension [72,73], and the simultaneous local metric dimension [16]. For some more specific information on several results on the local metric dimension of graphs we suggest the Ph.…”

“…Strong, local and adjacency variants of the simultaneous metric dimension have been introduced in [58], [16] and [152], respectively, in a natural way. We remark that concerning the simultaneous strong metric dimension of graph families, it was introduced in [190] the simultaneous version of the strong resolving graph for graph families.…”

On the k-partition dimension of graphs

“…Such variations have become more or less known and popular in connection to their applicability or according to the number of challenge that have arisen from them. Among them we could remark resolving dominating sets [6], weak total resolving sets [7,28], independent resolving sets [9], local metric sets [1,2,34], strong resolving sets [14,30,33], simultaneous metric dimension [1,14,35,36,37], strong resolving partitions [43,44] and resolving partitions [10,11,22,25,38,39]. A generalization of this last variation will be the object of study of this article.…”

“…If G is a connected graph of order n ≥ 2, then pd 2 (G) = n if and only if G ∼ = K n or G ∼ = K n − e.Proof. Suppose that pd 1 (G) = n. By Theorem 20, pd1 (G) = n if and only if d * (G) ≤ 2. Since 2 ≤ d(G) ≤ d * (G) ≤ 2, it follows that d(G) = d * (G) = 2.…”

Removing Twins in Graphs to Break Symmetries

“…These sets are a generalization of resolving sets, independently introduced by Slater [5] and Harary and Melter [6], motivated by the problem of identifying the location of an intruder in a network, by means of distances. Resolving sets and some related sets were recently studied in [7][8][9][10][11][12]. Determining sets and resolving sets were jointly studied (see [13,14]).…”