Optimal Local Identifying and Local Locating-dominating Codes

Issue title: Codes, Graphs, Coverings, and Identification: Special Issue Honoring the 60-th Birthday of Professor Iiro Honkala

Guest editors: Vesa Halava, Jarkko Kari and Tero Laihonen

Article type: Research Article

Authors: Herva, Pyry^{; *; †} | Laihonen, Tero^{; ‡} | Lehtilä, Tuomo^{; ‡}

Affiliations: Department of Mathematics and Statistics, University of Turku, Turku FI-20014, Finland. pysahe@utu.fi, terolai@utu.fi, tualeh@utu.fi

Correspondence: [*] Address for correspondence: Department of Mathematics and Statistics, University of Turku, Turku, FI-20014, Finland.

Note: [†] Research supported by the Emil Aaltonen Foundation.

Note: [‡] Research supported by the Academy of Finland grant 338797.

Abstract: We introduce two new classes of covering codes in graphs for every positive integer r. These new codes are called local r-identifying and local r-locating-dominating codes and they are derived from r-identifying and r-locating-dominating codes, respectively. We study the sizes of optimal local 1-identifying codes in binary hypercubes. We obtain lower and upper bounds that are asymptotically tight. Together the bounds show that the cost of changing covering codes into local 1-identifying codes is negligible. For some small n optimal constructions are obtained. Moreover, the upper bound is obtained by a linear code construction. Also, we study the densities of optimal local 1-identifying codes and local 1-locating-dominating codes in the infinite square grid, the hexagonal grid, the triangular grid and the king grid. We prove that seven out of eight of our constructions have optimal densities.

Keywords: Local identifying codes, local locating-dominating codes, identifying codes, locating-dominating codes, dominating sets, hypercubes, infinite grids, discharging methods

DOI: 10.3233/FI-242187

Journal: Fundamenta Informaticae, vol. 191, no. 3-4, pp. 351-378, 2024