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arXiv · 1504.00492

The Simultaneous Metric Dimension of Families Composed by Lexicographic Product Graphs

Yunior Ramirez-Cruz·Alejandro Estrada-Moreno·Juan A. Rodriguez-Velazquez
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Abstract

Let ${\mathcal G}$ be a graph family defined on a common (labeled) vertex set $V$. A set $S\subseteq V$ is said to be a simultaneous metric generator for ${\cal G}$ if for every $G\in {\cal G}$ and every pair of different vertices $u,v\in V$ there exists $s\in S$ such that $d_{G}(s,u)\ne d_{G}(s,v)$, where $d_{G}$ denotes the geodesic distance. A simultaneous adjacency generator for ${\cal G}$ is a simultaneous metric generator under the metric $d_{G,2}(x,y)=\min\{d_{G}(x,y),2\}$. A minimum cardinality simultaneous metric (adjacency) generator for ${\cal G}$ is a simultaneous metric (adjacency) basis, and its cardinality the simultaneous metric (adjacency) dimension of ${\cal G}$. Based on the simultaneous adjacency dimension, we study the simultaneous metric dimension of families composed by lexicographic product graphs.

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Yunior Ramirez-Cruz, Alejandro Estrada-Moreno, Juan A. Rodriguez-Velazquez. 2015-04-02. The Simultaneous Metric Dimension of Families Composed by Lexicographic Product Graphs. https://doi.org/10.1007/s00373-016-1675-1

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