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Three Problems on Well-Partitioned Chordal Graphs

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Algorithms and Complexity (CIAC 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12701))

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Abstract

In this work, we solve three problems on well-partitioned chordal graphs. First, we show that every connected (resp., 2-connected) well-partitioned chordal graph has a vertex that intersects all longest paths (resp., longest cycles). It is an open problem [Balister et al., Comb. Probab. Comput. 2004] whether the same holds for chordal graphs. Similarly, we show that every connected well-partitioned chordal graph admits a (polynomial-time constructible) tree 3-spanner, while the complexity status of the Tree 3-Spanner problem remains open on chordal graphs [Brandstädt et al., Theor. Comput. Sci. 2004]. Finally, we show that the problem of finding a minimum-size geodetic set is polynomial-time solvable on well-partitioned chordal graphs. This is the first example of a problem that is \(\mathsf {NP}\)-hard on chordal graphs and polynomial-time solvable on well-partitioned chordal graphs. Altogether, these results reinforce the significance of this recently defined graph class as a tool to tackle problems that are hard or unsolved on chordal graphs.

J. A. and O. K. are supported by the Institute for Basic Science (IBS-R029-C1). O. K. is also supported by the National Research Foundation of Korea (NRF) grant funded by the Ministry of Education (No. NRF-2018R1D1A1B07050294).

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Notes

  1. 1.

    The Helly property of trees states that in every tree, every collection of pairwise intersecting subtrees has a common nonempty intersection.

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Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, KAIST, Daejeon, South Korea

    Jungho Ahn

  2. Discrete Mathematics Group, IBS, Daejeon, South Korea

    Jungho Ahn & O-joung Kwon

  3. Department of Informatics, University of Bergen, Bergen, Norway

    Lars Jaffke & Paloma T. Lima

  4. Department of Mathematics, Incheon National University, Incheon, South Korea

    O-joung Kwon

Authors
  1. Jungho Ahn
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  2. Lars Jaffke
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  3. O-joung Kwon
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  4. Paloma T. Lima
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Correspondence to Lars Jaffke .

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Editors and Affiliations

  1. Sapienza University of Rome, Rome, Italy

    Tiziana Calamoneri

  2. Sapienza University of Rome, Rome, Italy

    Federico Corò

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Ahn, J., Jaffke, L., Kwon, Oj., Lima, P.T. (2021). Three Problems on Well-Partitioned Chordal Graphs. In: Calamoneri, T., Corò, F. (eds) Algorithms and Complexity. CIAC 2021. Lecture Notes in Computer Science(), vol 12701. Springer, Cham. https://doi.org/10.1007/978-3-030-75242-2_2

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  • DOI: https://doi.org/10.1007/978-3-030-75242-2_2

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