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The Simultaneous Local Metric Dimension of Graph Families

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Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, Av. Països Catalans 26, 43007 Tarragona, Spain
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Interdisciplinary Centre for Security, Reliability and Trust, University of Luxembourg, 6 av. de la Fonte, L-4364 Esch-sur-Alzette, Luxembourg
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Author to whom correspondence should be addressed.
Symmetry 2017, 9(8), 132; https://doi.org/10.3390/sym9080132
Received: 10 May 2017 / Revised: 21 July 2017 / Accepted: 24 July 2017 / Published: 27 July 2017
(This article belongs to the Special Issue Graph Theory)
In a graph $G = ( V , E )$ , a vertex $v ∈ V$ is said to distinguish two vertices x and y if $d G ( v , x ) ≠ d G ( v , y )$ . 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. View Full-Text
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Barragán-Ramírez, G.A.; Estrada-Moreno, A.; Ramírez-Cruz, Y.; Rodríguez-Velázquez, J. The Simultaneous Local Metric Dimension of Graph Families. Symmetry 2017, 9, 132.

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