Optimal local identifying and local locating-dominating codes

TL;DR

Together the bounds show that the cost of changing covering codes into local 1-identifying codes is negligible, and it is proved that seven out of eight of the authors' constructions have optimal densities.

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.

Figures (7)

• Figure 1: A graph that admits a local 2-identifying code but does not admit any 2-identifying codes. The darkened vertices form a local 2-identifying code.
• Figure 2: Illustration of the hierarchy between different classes of covering codes.
• Figure 3: The darkened vertices form a (non-optimal) covering code.
• Figure 4: Infinite grids.
• Figure 5: Local location-domination in the square and hexagonal grids.

Theorems & Definitions (35)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Lemma 1.5
• Corollary 1.6
• Corollary 1.7
• Theorem 1.8
• Definition 1.9
• Lemma 1.10