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Self-complementary (Pseudo-)Split Graphs

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LATIN 2024: Theoretical Informatics (LATIN 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14579))

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Abstract

We study split graphs and pseudo-split graphs that are isomorphic to their complements. These special subclasses of self-complementary graphs are actually the core of self-complementary graphs. Indeed, we show that all realizations of forcibly self-complementary degree sequences are pseudo-split graphs. We also give formulas to calculate the number of self-complementary (pseudo-)split graphs of a given order, and show that Trotignon’s conjecture holds for all self-complementary split graphs.

Supported by RGC grant 15221420 and NSFC grants 61972330 and 62372394.

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Notes

  1. 1.

    Some authors call such graph classes “self-complementary,” e.g., graphclasses.org.

  2. 2.

    Proofs of statements marked with \(\star \) can be found in the full version (arXiv:2312.10413).

  3. 3.

    The reader familiar with threshold graphs may note its use here. If we contract \(K_{i}\) and \(I_{i}\) into two vertices, the graph we constructed is a threshold graph. Threshold graphs have a stronger characterization by degree sequences. Since a threshold graph free of \(2K_{2}\), \(P_{4}\), and \(C_{4}\), no 2-switch is possible on it. Thus, the degree sequence of a threshold graph has a unique realization.

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Authors and Affiliations

  1. Department of Computing, Hong Kong Polytechnic University, Hong Kong, China

    Yixin Cao, Haowei Chen & Shenghua Wang

Authors
  1. Yixin Cao
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  2. Haowei Chen
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  3. Shenghua Wang
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Correspondence to Yixin Cao .

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Editors and Affiliations

  1. DIM-CMM, Universidad de Chile, Santiago, Chile

    José A. Soto

  2. Technical University of Munich, Munich, Germany

    Andreas Wiese

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Cao, Y., Chen, H., Wang, S. (2024). Self-complementary (Pseudo-)Split Graphs. In: Soto, J.A., Wiese, A. (eds) LATIN 2024: Theoretical Informatics. LATIN 2024. Lecture Notes in Computer Science, vol 14579. Springer, Cham. https://doi.org/10.1007/978-3-031-55601-2_1

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  • DOI: https://doi.org/10.1007/978-3-031-55601-2_1

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-55600-5

  • Online ISBN: 978-3-031-55601-2

  • eBook Packages: Computer ScienceComputer Science (R0)

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