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A polynomial-time exact algorithm for counting the number of directed acyclicgraphs in a Markov equivalence class was recently given by Wien\"obst, Bannach,and Li\'skiewicz (AAAI 2021). In this paper, we consider the more generalproblem of counting the number of directed acyclic graphs in a Markovequivalence class when the directions of some of the edges are also fixed (thissetting arises, for example, when interventional data is partially available).This problem has been shown in earlier work to be complexity-theoreticallyhard. In contrast, we show that the problem is nevertheless tractable in aninteresting class of instances, by establishing that it is ``fixed-parametertractable''. In particular, our counting algorithm runs in time that is boundedby a polynomial in the size of the graph, where the degree of the polynomialdoes \emph{not} depend upon the number of additional edges provided as input.

Counting Markov Equivalent Directed Acyclic Graphs Consistent with Background Knowledge