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2-Nested Matrices: Towards Understanding the Structure of Circle Graphs

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Abstract

A (0, 1)-matrix has the consecutive-ones property (C1P) if its columns can be permuted to make the 1’s in each row appear consecutively. This property was characterized in terms of forbidden submatrices by Tucker in 1972. Several graph classes were characterized by means of this property, including interval graphs and strongly chordal digraphs. In this work, we define and characterize 2-nested matrices, which are (0, 1)-matrices with a variant of the C1P and for which there is also a certain assignment of one of two colors to each block of consecutive 1’s in each row. The characterization of 2-nested matrices in the present work is of key importance to characterize split graphs that are also circle by minimal forbidden induced subgraphs.

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Acknowledgements

We would like to thank Luciano Grippo for his engagement during the early stages of this work.

Funding

This work was partially supported by ANPCyT PICT-2015-2218, UBACyT Grants 20020170100495BA and 20020160100095BA, Programa Regional MATHAMSUD MATH190013. Guillermo Durán partially supported by ISCI CONICYT PIA FB0816; ICM P-05-004-F (Chile). Nina Pardal partially supported by a CONICET doctoral scholarship. Martín D. Safe partially supported by Universidad Nacional del Sur Grants PGI 24/L103 and PGI 24/L115, and ANPCyT PICT-2017-1315.

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Authors and Affiliations

  1. CONICET-Universidad de Buenos Aires, Instituto de Cálculo (IC), Buenos Aires, Argentina

    Guillermo Durán & Nina Pardal

  2. Universidad de Buenos Aires, Facultad de Ciencias Exactas y Naturales, Departamento de Matemática, Buenos Aires, Argentina

    Guillermo Durán

  3. Departamento de Ingeniería Industrial, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Santiago, Chile

    Guillermo Durán

  4. CONICET-Universidad de Buenos Aires, Instituto de Investigación en Ciencias de la Computación (ICC), Buenos Aires, Argentina

    Nina Pardal

  5. Departamento de Matemática, Universidad Nacional del Sur (UNS), Bahía Blanca, Argentina

    Martín D. Safe

  6. INMABB, Universidad Nacional del Sur (UNS)-CONICET, Bahía Blanca, Argentina

    Martín D. Safe

Authors
  1. Guillermo Durán
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  2. Nina Pardal
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  3. Martín D. Safe
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Correspondence to Nina Pardal.

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Durán, G., Pardal, N. & Safe, M.D. 2-Nested Matrices: Towards Understanding the Structure of Circle Graphs. Graphs and Combinatorics 38, 111 (2022). https://doi.org/10.1007/s00373-022-02510-1

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  • DOI: https://doi.org/10.1007/s00373-022-02510-1

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