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Beyond Helly Graphs: The Diameter Problem on Absolute Retracts

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Graph-Theoretic Concepts in Computer Science (WG 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12911))

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Abstract

A subgraph H of a graph G is called a retract of G if it is the image of some idempotent endomorphism of G. We say that H is an absolute retract of some graph class \(\mathcal{C}\) if it is a retract of any \(G \in \mathcal{C}\) of which it is an isochromatic and isometric subgraph. In this paper, we study the complexity of computing the diameter within the absolute retracts of various hereditary graph classes. First, we show how to compute the diameter within absolute retracts of bipartite graphs in randomized \(\tilde{\mathcal{O}}(m\sqrt{n})\) time. Even on the proper subclass of cube-free modular graphs it is, to our best knowledge, the first subquadratic-time algorithm for diameter computation. For the special case of chordal bipartite graphs, it can be improved to linear time, and the algorithm even computes all the eccentricities. Then, we generalize these results to the absolute retracts of k-chromatic graphs, for every \(k \ge 3\). Finally, we study the diameter problem within the absolute retracts of planar graphs and split graphs.

This work was supported by project PN-19-37-04-01 “New solutions for complex problems in current ICT research fields based on modelling and optimization”, funded by the Romanian Core Program of the Ministry of Research and Innovation (MCI) 2019–2022.

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Notes

  1. 1.

    A vertex v is covered by another vertex w if \(N_G(v) \subseteq N_G(w)\) (a covered vertex is called embeddable in [68]).

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Authors and Affiliations

  1. University of Bucharest and National Institute for Research and Development in Informatics, Bucureşti, Romania

    Guillaume Ducoffe

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  1. Guillaume Ducoffe
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Correspondence to Guillaume Ducoffe .

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  1. University of Warsaw, Warsaw, Poland

    Łukasz Kowalik

  2. University of Warsaw, Warsaw, Poland

    Michał Pilipczuk

  3. Warsaw University of Technology, Warsaw, Poland

    Paweł Rzążewski

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Ducoffe, G. (2021). Beyond Helly Graphs: The Diameter Problem on Absolute Retracts. In: Kowalik, Ł., Pilipczuk, M., Rzążewski, P. (eds) Graph-Theoretic Concepts in Computer Science. WG 2021. Lecture Notes in Computer Science(), vol 12911. Springer, Cham. https://doi.org/10.1007/978-3-030-86838-3_25

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