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 \(\mathcal{G}\) if for every \(G\in \mathcal{G}\) 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 \(\mathcal{G}\) 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 \(\mathcal{G}\) is a simultaneous metric (adjacency) basis, and its cardinality the simultaneous metric (adjacency) dimension of \(\mathcal{G}\). Based on the simultaneous adjacency dimension, we study the simultaneous metric dimension of families composed by lexicographic product graphs.
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Notes
Adjacency generators were called adjacency resolving sets in [12].
For any pair of vertices x, y belonging to different connected components of G we can assume that \(d_G(x,y)=\infty \) and so \(d_{G,t}(x,y)=t\) for any t greater than or equal to the maximum diameter of a connected component of G.
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Ramírez-Cruz, Y., Estrada-Moreno, A. & Rodríguez-Velázquez, J.A. The Simultaneous Metric Dimension of Families Composed by Lexicographic Product Graphs. Graphs and Combinatorics 32, 2093–2120 (2016). https://doi.org/10.1007/s00373-016-1675-1
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DOI: https://doi.org/10.1007/s00373-016-1675-1