# Question about Complete partially Directed Acyclic Graph

I am reading CPDAG, can anyone please explain why G1 and G3 are not equivalence classes as in the picture below? Thank you very much!

"Theorem 1 [59] Two DAGs are equivalent if and only if they have the same skeleton and the same v-structures.

The skeleton of a DAG is the undirected graph that results from ignoring the directionality of every edge.

A v-structure in a DAG $H$ is an ordered triplet of nodes, $(x,y,z)$, such that (1) $H$ contains the arcs $x\rightarrow y$ and $y \leftarrow z$, and (2) the nodes $x$ and $z$ are not adjacent in $H$. A head-to-head pattern (shortened h-h) in a DAG $H$ is an ordered triplet of nodes, $(x,y,z)$, such that $H$ contains the arcs $x\rightarrow y$ and $y \leftarrow z$. Note that in an h-h pattern $(x,y,z)$ the nodes $x$ and $z$ can be adjacent."

The concept of equivalence of DAGs partitions the space of DAGs into a set of equivalence classes.

• Could you please explain/define the term CPDAG in your post? Also what do you think yourself, why aren't G1 and G3 in equivalence class, any idea? – fade2black Aug 13 '17 at 11:17
• It's also weird why you ask "why $G_1$ is not in class $G$" when I can read on the fine prints under the picture saying $G_1$ is in class $G$. – Billiska Aug 13 '17 at 12:04
• Thank you. I added the description in the question. Please have a look. Thank you! – user2842390 Aug 13 '17 at 14:55
• What is a CPDAG? (You've expanded the acronym but not given a definition.) What does "equivalent" mean? – David Richerby Jan 11 '18 at 1:25

They have different v-structures: $G_1$ has the v-structures $(x_3,y,x_4),(x_4,y,x_3)$, whereas $G_3$ has the v-structures $(x_1,x_4,y),(y,x_4,x_1)$.
• Thank you, however, I can say so with G1 and G4 (but the example said that G1 G4 are in its equivalence class): "$G_1$ has the v-structures $(x_3,y,x_4),(x_4,y,x_3)$ whereas $G_4$ has the v-structures $(x_1,x_4,y),(y,x_4,x_1)$". So why are they sill equivalent? – user2842390 Aug 14 '17 at 6:25