Counting Markov Equivalence Classes by Number of Immoralities

Abstract

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