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Parallelogram Morphisms and Circular Codes

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Language and Automata Theory and Applications (LATA 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9618))

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Abstract

In 2014, it was conjectured that any polyomino can be factorized uniquely as a product of prime polyominoes [7]. In this paper, we present simple tools from words combinatorics and graph topology that seem very useful in solving the conjecture. The main one is called parallelogram network, which is a particular subgraph of \(G(\mathbb {Z}^2)\) induced by a parallelogram morphism, i.e. a morphism describing the contour of a polyomino tiling the plane as a parallelogram would. In particular, we show that parallelogram networks are homeomorphic to \(G(\mathbb {Z}^2)\). This leads us to show that the image of the letters of parallelogram morphisms is a circular code provided each element is primitive, therefore solving positively a 2013 conjecture [8].

H. Tremblay—This research is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).

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Authors and Affiliations

  1. Laboratoire de Combinatoire et d’informatique Mathématique, Université du Québec à Montréal, CP 8888, Succ. Centre-ville, Montréal, Québec, H3C 3P8, Canada

    Alexandre Blondin Massé, Mélodie Lapointe & Hugo Tremblay

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  1. Alexandre Blondin Massé
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  2. Mélodie Lapointe
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  3. Hugo Tremblay
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Correspondence to Hugo Tremblay .

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Editors and Affiliations

  1. Rovira i Virgili University, Tarragona, Spain

    Adrian-Horia Dediu

  2. Czech Technical University, Prague, Czech Republic

    Jan Janoušek

  3. Rovira i Virgili University, Tarragona, Spain

    Carlos Martín-Vide

  4. Justus Liebig University Giessen, Gießen, Germany

    Bianca Truthe

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Massé, A.B., Lapointe, M., Tremblay, H. (2016). Parallelogram Morphisms and Circular Codes. In: Dediu, AH., Janoušek, J., Martín-Vide, C., Truthe, B. (eds) Language and Automata Theory and Applications. LATA 2016. Lecture Notes in Computer Science(), vol 9618. Springer, Cham. https://doi.org/10.1007/978-3-319-30000-9_17

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  • DOI: https://doi.org/10.1007/978-3-319-30000-9_17

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-29999-0

  • Online ISBN: 978-3-319-30000-9

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