The Simultaneous Metric Dimension of Graph Families
Y. Ramirez-Cruz, O. R. Oellermann, J. A. Rodriguez-Velazquez

TL;DR
This paper investigates the concept of simultaneous metric dimension in graph families, providing bounds, formulas, and complexity results, especially for trees, to understand how to efficiently determine resolving sets across multiple graphs.
Contribution
It introduces the simultaneous metric dimension, derives bounds and formulas for various graph families, and analyzes the computational complexity for trees.
Findings
Sharp bounds for general graph families.
NP-hardness of the problem for trees.
Exact bounds for specific tree transformations.
Abstract
A vertex is said to resolve two vertices and if . A set is said to be a metric generator for if any pair of vertices of is resolved by some element of . A minimum metric generator is called a metric basis, and its cardinality, , the \emph{metric dimension} of . A set is said to be a simultaneous metric generator for a graph family , defined on a common (labeled) vertex set, if it is a metric generator for every graph of the family. A minimum cardinality simultaneous metric generator is called a simultaneous metric basis, and its cardinality the simultaneous metric dimension of . We obtain sharp bounds for this invariants for general families of graphs and calculate closed formulae or tight bounds for the simultaneous metric dimension of several specific…
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