Exact values for three domination-like problems in circular and infinite grid graphs of small height

TL;DR

This paper determines exact minimum sizes for identifying, locating-dominating, and locating-total-dominating codes in small-height circular and infinite grid graphs, using a computer-assisted search method.

Contribution

It offers new exact values for these codes and introduces a generic computational framework for such problems in grid graphs.

Findings

Exact minimum code sizes for specific grid graphs are provided.

A novel computer search framework for domination-like problems is developed.

The paper confirms and extends previous results with new proofs and data.

Abstract

In this paper we study three domination-like problems, namely identifying codes, locating-dominating codes, and locating-total-dominating codes. We are interested in finding the minimum cardinality of such codes in circular and infinite grid graphs of given height. We provide an alternate proof for already known results, as well as new results. These were obtained by a computer search based on a generic framework, that we developed earlier, for the search of a minimum labeling satisfying a pseudo-d-local property in rotagraphs.