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Open Problems and Recent Developments on a Complexity Framework for Forbidden Subgraphs

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SOFSEM 2025: Theory and Practice of Computer Science (SOFSEM 2025)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 15538))

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Abstract

For any finite set \(\mathcal {H} = \{H_1,\ldots ,H_p\}\) of graphs, a graph is \(\mathcal {H}\)-subgraph-free if it does not contain any of \(H_1,\ldots ,H_p\) as a subgraph. In this invited talk, I discuss a recently proposed algorithmic meta classification that precisely classifies if certain problems (sharing specific properties) are “efficiently solvable” or “computationally hard” for \(\mathcal {H}\)-subgraph-free graphs, depending on \(\mathcal {H}\). For a broad set of classic graph problems, this framework yields a dichotomy (depending on \(\mathcal {H}\)) between polynomial-time solvability and NP-completeness. For other problems, like computing the diameter of a graph, it gives a dichotomy between almost-linear-time solvability and having no subquadratic-time algorithm (conditioned on some hardness hypotheses). This paper discusses this framework, highlights current developments and open problems, and surveys recent insights into the complexity on \(\mathcal {H}\)-subgraph-free graphs of problems that do not fall within the framework.

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Acknowledgments

I want to thank my coauthors of the works that motivated this invited talk: Hans Bodlaender, Matthew Johnson, Vadim Lozin, Barnaby Martin, Jelle Oostveen, Sukanya Pandey, Daniel Paulusma, Mark Siggers, and Siani Smith. I also want to thank Daniel Paulusma for helpful discussions on this paper and for suggesting Proposition 1.

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  1. Department Information and Computing Sciences, Utrecht University, Utrecht, The Netherlands

    Erik Jan van Leeuwen

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  1. Erik Jan van Leeuwen
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Correspondence to Erik Jan van Leeuwen .

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  1. Comenius University in Bratislava, Bratislava, Slovakia

    Rastislav Královič

  2. Czech Academy of Sciences, Prague, Czech Republic

    Věra Kůrková

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van Leeuwen, E.J. (2025). Open Problems and Recent Developments on a Complexity Framework for Forbidden Subgraphs. In: Královič, R., Kůrková, V. (eds) SOFSEM 2025: Theory and Practice of Computer Science. SOFSEM 2025. Lecture Notes in Computer Science, vol 15538. Springer, Cham. https://doi.org/10.1007/978-3-031-82670-2_2

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  • DOI: https://doi.org/10.1007/978-3-031-82670-2_2

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