Abstract
We consider a natural generalization of chordal graphs, in which every minimal separator induces a subgraph with independence number at most 2. Such graphs can be equivalently defined as graphs that do not contain the complete bipartite graph \(K_{2,3}\) as an induced minor, that is, graphs from which \(K_{2,3}\) cannot be obtained by a sequence of edge contractions and vertex deletions.
We develop a polynomial-time algorithm for recognizing these graphs. Our algorithm relies on a characterization of \(K_{2,3}\)-induced minor-free graphs in terms of excluding particular induced subgraphs, called Truemper configurations.
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Notes
- 1.
For a simpler example, consider the case when \(\mathcal F\) consists of all complements of cycles.
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Acknowledgments
The authors are grateful to Nevena Pivač for helpful discussions. This work is supported in part by the Slovenian Research and Innovation Agency (I0-0035, research program P1-0285 and research projects J1-3001, J1-3002, J1-3003, J1-4008, J1-4084, N1-0102, and N1-0160, and BI-FR/22-23-PROTEUS-01), by the research program CogniCom (0013103) at the University of Primorska, by the French Fédération de Recherche ICVL (Informatique Centre-Val de Loire), by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program Investissements d’Avenir (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR), by Agence Nationale de la Recherche (France) under research grant ANR DIGRAPHS ANR-19-CE48-0013-01 and by H2020-MSCA-RISE project CoSP-GA No. 823748.
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Dallard, C., Dumas, M., Hilaire, C., Milanič, M., Perez, A., Trotignon, N. (2024). Detecting \(K_{2,3}\) as an Induced Minor. In: Rescigno, A.A., Vaccaro, U. (eds) Combinatorial Algorithms. IWOCA 2024. Lecture Notes in Computer Science, vol 14764. Springer, Cham. https://doi.org/10.1007/978-3-031-63021-7_12
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