Refereed article in conference proceedings (A4)

On Perfect Coverings of Two-Dimensional Grids

List of Authors: Heikkilä Elias, Herva Pyry, Kari Jarkko

Conference name: International Conference on Developments in Language Theory

Place: Cham

Publication year: 2022

Journal: Lecture Notes in Computer Science

Book title *: Developments in Language Theory: 26th International Conference, DLT 2022, Tampa, FL, USA, May 9–13, 2022, Proceedings

Title of series: Lecture Notes in Computer Science

Volume number: 13257

Start page: 152

End page: 163

ISBN: 978-3-031-05577-5

eISBN: 978-3-031-05578-2

ISSN: 0302-9743

eISSN: 1611-3349

DOI: http://dx.doi.org/10.1007/978-3-031-05578-2_12

URL: https://link.springer.com/chapter/10.1007/978-3-031-05578-2_12

We study perfect multiple coverings in translation invariant graphs with vertex set Z2 using an algebraic approach. In this approach we consider any such covering as a two-dimensional binary configuration which we then express as a two-variate formal power series. Using known results, we conclude that any perfect multiple covering has a non-trivial periodizer, that is, there exists a non-zero polynomial whose formal product with the power series presenting the covering is a two-periodic configuration. If a non-trivial periodizer has line polynomial factors in at most one direction, then the configuration is known to be periodic. Using this result we find many setups where perfect multiple coverings of infinite grids are necessarily periodic. We also consider some algorithmic questions on finding perfect multiple coverings.