 
Summary: 1
The Dimensionality of Mixed Ancestral Graphs
by Peter Spirtes, Thomas Richardson and Chris Meek
First we will introduce some graph terminology. The concepts defined here are
illustrated in Figure 1. A graph consists of two parts, a set of vertices V and a set of edges
E. Each edge in E is between two distinct vertices in V. There are two kinds of edges in E,
directed edges A ® B or A ¬ B, and doubleheaded edges A « B; in either case A and B
are endpoints of the edge; further, A and B are said to be adjacent. In Figure 1 the set of
vertices is {A,B,C,D,E} and the set of edges is {A « B, B ® C, C ® D, E ® D}. For a
directed edge A ® B, A is the tail of the edge and B is the head of the edge, A is a
parent of B, and B is a child of A.
An undirected path U between X 1 and X n is a sequence of edges such
that one endpoint of E 1 is X 1 , one endpoint of E m is X n , and for each pair of consecutive
edges E i , E i+1 in the sequence, E i ¹ E i+1 , and one endpoint of E i equals one endpoint of E i+1 .
In Figure 1, A « B ® C ¬ D is an example of an undirected path between A and D. A
directed path P between X 1 and X n is a sequence of directed edges such that
the tail of E 1 is X 1 , the head of E m is X n , and for each pair of edges E i , E i+1 adjacent in the
sequence, E i ¹ E i+1 , and the head of E i is the tail of E i+1 . For example, B ® C ® D is a
directed path. A vertex occurs on a path if it is an endpoint of one of the edges in the
path. The set of vertices on A « B ® C ® D is {A, B, C, D}. A path is acyclic if no
