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Intersection Graphs of Non-crossing Paths

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Graph-Theoretic Concepts in Computer Science (WG 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11789))

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Abstract

We study graph classes modeled by families of non-crossing (NC) connected sets. Two classic graph classes in this context are disk graphs and proper interval graphs. We focus on the cases when the sets are paths and the host is a tree. Forbidden induced subgraph characterizations and linear time certifying recognition algorithms are given for intersection graphs of NC paths of a tree (and related subclasses). For intersection graphs of NC paths of a tree, the dominating set problem is shown to be solvable in linear time. Also, each such graph is shown to have a Hamiltonian cycle if and only if it is 2-connected, and to have a Hamiltonian path if and only if its block-cutpoint tree is a path.

The full version of this article with the appendix referred to herein is on arxiv.org [3].

S. Chaplick—Research supported by DFG grant WO 758/11-1.

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Notes

  1. 1.

    Note: all \(\exists \mathbb {R}\)-hard problems are NP-hard, see [28] for an introduction to \(\exists \mathbb {R}\).

  2. 2.

    Usually defined as having no interval strictly contained within any other.

  3. 3.

    A graph is a split graph when its vertices can be partitioned into a clique and an independent set. It is easy to see that split graphs are chordal.

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Authors and Affiliations

  1. Lehrstuhl für Informatik I, Universität Würzburg, Würzburg, Germany

    Steven Chaplick

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  1. CNRS, LIRMM, Université de Montpellier, Montpellier, France

    Ignasi Sau

  2. CNRS, LIRMM, Université de Montpellier, Montpellier, France

    Dimitrios M. Thilikos

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Chaplick, S. (2019). Intersection Graphs of Non-crossing Paths. In: Sau, I., Thilikos, D. (eds) Graph-Theoretic Concepts in Computer Science. WG 2019. Lecture Notes in Computer Science(), vol 11789. Springer, Cham. https://doi.org/10.1007/978-3-030-30786-8_24

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  • DOI: https://doi.org/10.1007/978-3-030-30786-8_24

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