Basically got graph transformation related to graph
isomorphism.

Define $G \to G'$.  $V(G')=V(G) \cup E(G)=\{v_1\ldots v_n\} \cup \{e_1\ldots e_m\}$. Call $v_i$ vertices $v'$ and $e_i$ vertices $e'$.

Edges of $G'$.

(1) Add $(v_i,e_j)$ iff $v_i \in e_j$.
This graph is bipartite and is the subdivision of $G$. According to a paper
this preserves GI.

(2) Make a clique of $v'$ vertices, i.e.
add $(v_i,v_j)$ for $i \ne j$ and without
multiple edges. This graph is chordal and
split and according to paper preserves GI.


(3) Make a clique of $e'$ vertices, i.e.
add $(e_i,e_j)$ for $i \ne j$ and without
multiple edges.

Vertices of $G'$ can be partitioned
into two cliques on $v',e'$ and
the edges between the cliques are from (1),
are the subdivision of $G$.

$G \cong H \implies G' \cong H'$.

> Q1 Does this transformation preserves isomorphism? $G' \cong H' \implies G \cong H$?

I am inclined to believe so, since the cliques doesn't matter
much and the subdivision subgraph of (1) preserves isomorphism.

A graph is $X$-free if it doesn't contain induced subgraph
$X$.

Claim 1. $G'$ is $(P_4 \cup K_1)$-free.

$3$ or more $v'$ vertices induce triangle and $3$ or more $e'$ vertices
induce triangle since they are in a clique.
The $5$ vertices of $(P_4 \cup K_1)$ can't be partitioned
in $v',e'$ without a triangle.

> Is this correct?

Conjecture based on experimental data and suggested proof:

If $G$ is triangle-free, $G'$ is $\overline{3K_2}$-free.
$\overline{3K_2}=K_{2,2,2}$ and the clique number is $3$,
so if induced $\overline{3K_2}$ would exists it must be on $3e' \cup 3v'$
vertices. I suspect that the $v'$ vertices must induce triangle in $G$
for the following reason: The partitions of $K_{2,2,2}=\overline{3K_2}$ are
$(v_1,e_1),(v_2,e_2),(v_3,e_3)$. The edges $(v_i,v_j),(e_i,e_j)$
come from the cliques. The remaining edges are 
$(v_1,\{e_2,e_3\}),(v_2,\{e_1,e_3\}),(v_3,\{e_1,e_2\})$,
which is $C_6$, subdivision of triangle, contradicting triangle-free.



Counterexamples are welcome.

For triangle-free $G$, $G'$ is $(P_4 \cup K_1,\overline{3K2})$-free.

Appears to me correctness of the above and
[paper top of p. 10](http://arxiv.org/abs/1208.0142)
would imply GI is in P.