Complete graphs have different v-structures? From the book "Probabilistic Graphical Models", here says two complete graphs have different v-structures. As I understand, v-structure is like "X->Z<-Y" without edge between X and Y. If so, complete graphs have no v-structures so they have the same v-structures. Hope someone helps me to make it clear.

 A: In Bayesian networks, complete graph definition is slightly different than usual (i.e. complete digraph). The graph is complete if every pair of nodes are connected by some edge and the graph is still acyclic. Therefore, as also noted in the book, any addition of an edge creates a cycle in the graph because an edge in the inverse direction already exists in the graph. 
Therefore, complete graphs are I-equivalent since $I(G)$ is empty but this isn't necessary for v-structures. You can play with the directions of edges and get a completely different set of v-structures. For example, the graph $\{X\rightarrow Y,X\rightarrow Z,Y\rightarrow Z\}$ has different v-structure (i.e. $X\rightarrow Z\leftarrow Y$) than the complete graph $\{Z\rightarrow Y,X\rightarrow Y, Z\rightarrow X\}$ with the v-structure $\{Z\rightarrow Y\leftarrow X\}$. Having an edge between parents doesn't violate the concept of v-structure (at least in your book, as @BenReiniger notes). It is rather relevant to immorality of the network. 
