Periodicity of identifying codes in strips

TL;DR

This paper proves that the lowest possible density of identifying codes in infinite strips of a grid can always be realized by periodic codes with bounded pattern length, and explicitly finds such codes for strips of 4 and 5 rows.

Contribution

It establishes the existence of periodic identifying codes with bounded pattern length in strips and computes minimal density codes for specific cases using algorithmic methods.

Findings

Minimal density in S_4 is 11/28.

Minimal density in S_5 is 19/50.

Periodic codes achieve the infimum density.

Abstract

An identifying code in a graph is a subset of vertices having a nonempty and distinct intersection with the closed neighborhood of every vertex. We prove that the infimum density of any identifying code in $S_k$ (an infinite strip of $k$ rows in the square grid) can always be achieved by a periodic identifying code with pattern length at most $2^{4k}$. Assisted by a compute program implementing Karp's algorithm for minimum cycle mean, we find a periodic identifying code in $S_4$ with the minimum density $11/28$, and a periodic identifying code in $S_5$ with the minimum density $19/50$.