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k-Leaf Powers Cannot Be Characterized by a Finite Set of Forbidden Induced Subgraphs for k ≥ 5

Authors Max Dupré la Tour, Manuel Lafond , Ndiamé Ndiaye , Adrian Vetta

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  • Mathematics of computing → Graph algorithms
Keywords
  • Leaf Powers
  • Forbidden Graph Characterizations
  • Strongly Chordal Graphs

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Abstract

A graph G = (V,E) is a k-leaf power if there is a tree T whose leaves are the vertices of G, with the property that a pair of distinct leaves u and v share an edge in G if and only if they are distance at most k apart in T. For k ≤ 4, it is known that there exists a finite set F_k of graphs such that the class ℒ(k) of k-leaf power graphs is characterized as the set of strongly chordal graphs that do not contain any graph in F_k as an induced subgraph. We prove no such characterization holds for k ≥ 5. That is, for any k ≥ 5, there is no finite set F_k of graphs such that ℒ(k) is equivalent to the set of strongly chordal graphs that do not contain as an induced subgraph any graph in F_k.

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Max Dupré la Tour, Manuel Lafond, Ndiamé Ndiaye, and Adrian Vetta. k-Leaf Powers Cannot Be Characterized by a Finite Set of Forbidden Induced Subgraphs for k ≥ 5. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 72:1-72:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.72

Author Details

Max Dupré la Tour
  • McGill University, Montreal, Canada
Manuel Lafond
  • Université de Sherbrooke, Canada
Ndiamé Ndiaye
  • McGill University, Montreal, Canada
Adrian Vetta
  • McGill University, Montreal, Canada

Acknowledgements

This research collaboration was instigated at the Montreal Graph Theory Workshop.

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