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Two directed acyclic graphs (DAGs) are called Markov equivalent if and onlyif they have the same underlying undirected graph (i.e. skeleton) and the sameset of immoralities. Using observational data, a DAG model can only bedetermined up to Markov equivalence, and so it is desirable to understand thesize and number of Markov equivalence classes (MECs) combinatorially. In thispaper, we address this enumerative question using a pair of generatingfunctions that encode the number and size of MECs on a skeleton $G$, and indoing so we connect this problem to classical problems in combinatorialoptimization. The first is a graph polynomial that counts the number of MECs on$G$ by their number of immoralities. Using connections to the independent setproblem, we show that computing a DAG on $G$ with the maximum possible numberof immoralities is NP-hard. The second generating function counts the MECs on$G$ according to their size. Via computer enumeration, we show that thisgenerating function is distinct for every connected graph on $p$ nodes for all$p\leq 10$.

Counting Markov Equivalence Classes by Number of Immoralities