Here I try to seek if restricting the structure of permutations would still keep GI property.
Given a $2n$ vertex undirected graph whose vertices are partitioned arbitrarily in pairs to say WLOG $(1,2)$, $(3,4)$, $\dots$, $(2n-1,2n)$. Call these vertices pairs as super vertices.
Call two such graphs $2n$ vertex labelled graphs $G$, $H$ whose adjacencies are $A$ and $B$ respectively $\mathsf{vertex}$-$\mathsf{pair}$-$\mathsf{isomorphic}$ iff
($\star$) there is a permutation $P$ such that $A=PBP'$ on the condition that the permutation $P$ permutes only supervertices.
Note that only a subset $n!$ of permutations of $(2n)!$ allowed.
In particular,
($\star$) vertex $2i$ in $G$ is mapped to vertex $2j$ in $H$ by the permutation iff vertex $2i-1$ in $G$ must be mapped to $2j-1$ in $H$.
Is $\mathsf{vertex}$-$\mathsf{pair}$-$\mathsf{isomorphic}$ $\mathsf{GI}$-$\mathsf{complete}$?
Also posted: http://cs.stackexchange.com/questions/53824/restricting-possible-permutations-in-graph-isomorphism-problemhttps://cs.stackexchange.com/questions/53824/restricting-possible-permutations-in-graph-isomorphism-problem