Counting and Sampling Labeled Chordal Graphs in Polynomial Time

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 omega as input, it computes the number of omega-colorable labeled chordal graphs on n vertices, using O(n^7) arithmetic operations. A standard sampling-to-counting reduction then yields a polynomial-time exact sampler that generates an omega-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.