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arXiv · 2501.05024

Sampling Unlabeled Chordal Graphs in Expected Polynomial Time

Úrsula Hébert-Johnson·Daniel Lokshtanov
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Abstract

We design an algorithm that generates an $n$-vertex unlabeled chordal graph uniformly at random in expected polynomial time. Along the way, we develop the following two results: (1) an $\mathsf{FPT}$ algorithm for counting and sampling labeled chordal graphs with a given automorphism $\pi$, parameterized by the number of moved points of $\pi$, and (2) a proof that the probability that a random $n$-vertex labeled chordal graph has a given automorphism $\pi\in S_n$ is at most $1/2^{c\max\{\mu^2,n\}}$, where $\mu$ is the number of moved points of $\pi$ and $c$ is a constant. Our algorithm for sampling unlabeled chordal graphs calls the aforementioned $\mathsf{FPT}$ algorithm as a black box with potentially large values of the parameter $\mu$, but the probability of calling this algorithm with a large value of $\mu$ is exponentially small.

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Úrsula Hébert-Johnson, Daniel Lokshtanov. 2025-01-09. Sampling Unlabeled Chordal Graphs in Expected Polynomial Time. https://arxiv.org/abs/2501.05024

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