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On Models of Directed Path Graphs Non Rooted Directed Path Graphs

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Abstract

A directed path graph is the intersection graph of a family of directed subpaths of a directed tree. A rooted directed path graph is the intersection graph of a family of directed subpaths of a rooted tree. Clearly, rooted directed path graphs are directed path graphs. Several characterizations are known for directed path graphs: one by forbidden induced subgraphs and one by forbidden asteroids. It is an open problem to find such characterizations for rooted directed path graphs. With the purpose of proving knowledge in this direction, we show in this paper properties of directed path models that can not be rooted for chordal graphs with any leafage and with leafage four. Therefore, we prove that for leafage four directed path graphs minimally non rooted directed path graphs have a unique asteroidal quadruple, and can be characterized by the presence of certain type of asteroidal quadruples.

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Acknowledgments

We thank referees for their comments and suggestions which improved the presentation of the earlier draft of this article.

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Authors and Affiliations

  1. Conicet, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, Calle 50 y 115, 1900, La Plata, Argentina

    Marisa Gutierrez

  2. Departamento de Matemática, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, Calle 50 y 115, 1900, La Plata, Argentina

    Marisa Gutierrez & Silvia B. Tondato

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  1. Marisa Gutierrez
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  2. Silvia B. Tondato
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Correspondence to Silvia B. Tondato.

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Gutierrez, M., Tondato, S.B. On Models of Directed Path Graphs Non Rooted Directed Path Graphs. Graphs and Combinatorics 32, 663–684 (2016). https://doi.org/10.1007/s00373-015-1600-z

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  • DOI: https://doi.org/10.1007/s00373-015-1600-z

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