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Counting and Sampling Labeled Chordal Graphs in Polynomial Time

Authors Úrsula Hébert-Johnson , Daniel Lokshtanov, Eric Vigoda

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Úrsula Hébert-Johnson
  • University of California, Santa Barbara, CA, USA
Daniel Lokshtanov
  • University of California, Santa Barbara, CA, USA
Eric Vigoda
  • University of California, Santa Barbara, CA, USA

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Úrsula Hébert-Johnson, Daniel Lokshtanov, and Eric Vigoda. Counting and Sampling Labeled Chordal Graphs in Polynomial Time. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 58:1-58:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


We present the first polynomial-time algorithm to exactly compute the number of labeled chordal graphs on n vertices. Our algorithm solves a more general problem: given n and ω as input, it computes the number of ω-colorable labeled chordal graphs on n vertices, using O(n⁷) arithmetic operations. A standard sampling-to-counting reduction then yields a polynomial-time exact sampler that generates an ω-colorable labeled chordal graph on n vertices uniformly at random. Our counting algorithm improves upon the previous best result by Wormald (1985), which computes the number of labeled chordal graphs on n vertices in time exponential in n. An implementation of the polynomial-time counting algorithm gives the number of labeled chordal graphs on up to 30 vertices in less than three minutes on a standard desktop computer. Previously, the number of labeled chordal graphs was only known for graphs on up to 15 vertices.

Subject Classification

ACM Subject Classification
  • Theory of computation → Generating random combinatorial structures
  • Theory of computation → Graph algorithms analysis
  • Counting algorithms
  • graph sampling
  • chordal graphs


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