For digraphs $G$ and $H$ if we can partition $V(G)$ into a family $\{Q\_t\}\_{t\in V(H)}$ indexed by $V(H)$ such that $E(G)=\bigcup\_{(u,v)\in E(H)}Q\_u\times Q\_v$, then is every subgraph of $G$ isomorphic to $H$ induced in $G$ by some set of complete representatives for the partition $P=\{Q\_t:t\in V(H)\}$?



Enumerating isomorphic subgraphs
This is not true in general. Here is a non-trivial counter example: Let $H \sim K\_{1,2}$ and $G\sim C\_4$, with the vertices of $G$ labelled $a,b,c,d$ around the cycle. Then $\{a\},\{c\},\{b,d\}$ works as a partition, but the induced subgraph $G[a,b,d]$ is isomorphic to $H$.


