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Linear Time Algorithms for NP-Hard Problems Restricted to GaTEx Graphs

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Computing and Combinatorics (COCOON 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14422))

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Abstract

The class of Galled-Tree Explainable (GaTEx) graphs has just recently been discovered as a natural generalization of cographs. Cographs are precisely those graphs that can be uniquely represented by a rooted tree where the leaves of the tree correspond to the vertices of the graph. As a generalization, GaTEx graphs are precisely those graphs that can be uniquely represented by a particular rooted directed acyclic graph (called galled-tree).

We consider here four prominent problems that are, in general, NP-hard: computing the size \(\omega (G)\) of a maximum clique, the size \(\chi (G)\) of an optimal vertex-coloring and the size \(\alpha (G)\) of a maximum independent set of a given graph G as well as determining whether a graph is perfectly orderable. We show here that \(\omega (G)\), \(\chi (G)\), \(\alpha (G)\) can be computed in linear-time for GaTEx graphs G. The crucial idea for the linear-time algorithms is to avoid working on the GaTEx graphs G directly, but to use the galled-trees that explain G as a guide for the algorithms to compute these invariants. In particular, we show first how to employ the galled-tree structure to compute a perfect ordering of GaTEx graphs in linear-time which is then used to determine \(\omega (G)\), \(\chi (G)\), \(\alpha (G)\).

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Stockholm University, 10691, Stockholm, Sweden

    Marc Hellmuth

  2. Bioinformatics Group, Department of Computer Science and Interdisciplinary Center for Bioinformatics, Universität Leipzig, 04107, Leipzig, Germany

    Guillaume E. Scholz

Authors
  1. Marc Hellmuth
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  2. Guillaume E. Scholz
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Correspondence to Guillaume E. Scholz .

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Editors and Affiliations

  1. University of Texas at Dallas, Richardson, TX, USA

    Weili Wu

  2. University of Delaware, Newark, DE, USA

    Guangmo Tong

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Hellmuth, M., Scholz, G.E. (2024). Linear Time Algorithms for NP-Hard Problems Restricted to GaTEx Graphs. In: Wu, W., Tong, G. (eds) Computing and Combinatorics. COCOON 2023. Lecture Notes in Computer Science, vol 14422. Springer, Cham. https://doi.org/10.1007/978-3-031-49190-0_8

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  • DOI: https://doi.org/10.1007/978-3-031-49190-0_8

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-49189-4

  • Online ISBN: 978-3-031-49190-0

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