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We present the first polynomial-time algorithm to exactly compute the numberof labeled chordal graphs on n vertices. Our algorithm solves a more generalproblem: given n and omega as input, it computes the number of omega-colorablelabeled chordal graphs on n vertices, using O(n^7) arithmetic operations. Astandard sampling-to-counting reduction then yields a polynomial-time exactsampler that generates an omega-colorable labeled chordal graph on n verticesuniformly at random. Our counting algorithm improves upon the previous bestresult by Wormald (1985), which computes the number of labeled chordal graphson n vertices in time exponential in n.  An implementation of the polynomial-time counting algorithm gives the numberof labeled chordal graphs on up to 30 vertices in less than three minutes on astandard desktop computer. Previously, the number of labeled chordal graphs wasonly known for graphs on up to 15 vertices.

Counting and Sampling Labeled Chordal Graphs in Polynomial Time