Minimum-Density Identifying Codes in Square Grids

  • Marwane Bouznif
  • Frédéric Havet
  • Myriam Preissmann
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9778)

Abstract

An identifying code in a graph G is a subset of vertices with the property that for each vertex \(v \in V(G)\), the collection of elements of C at distance at most 1 from v is non-empty and distinct from the collection of any other vertex. We consider the minimum density \(d^*(\mathcal{S}_k)\) of an identifying code in the square grid \(\mathcal{S}_k\) of height k (i.e. with vertex set \( \mathbb {Z} \times \{1, \dots , k\}\)). Using the Discharging Method, we prove \(\displaystyle \frac{7}{20} + \frac{1}{20k} \le d^*(\mathcal{S}_k) \le \min \left\{ \frac{2}{5}, \frac{7}{20} + \frac{3}{10k} \right\} \), and \(\displaystyle d^*(\mathcal{S}_3) =\frac{7}{18}\).

Keywords

Identifying code Square grid Discharging method 

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Marwane Bouznif
    • 1
  • Frédéric Havet
    • 2
  • Myriam Preissmann
    • 3
  1. 1.A-SISSaint ÉtienneFrance
  2. 2.Projet COATI, I3S (CNRS, UNSA) and INRIASophia AntipolisFrance
  3. 3.Univ. Grenoble Alpes and CNRS, G-SCOPGrenobleFrance

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