What data structure does Graph Isomorphism Problem assume?

Question

In Haskell, I can easily think of a data structure of quasi-labeled simple graphs:

{-# LANGUAGE DataKinds #-}
{-# LANGUAGE KindSignatures #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE GADTs #-}

data Graph (n :: Nat) where
    G1 :: Graph 1
    (:-) :: Int -> Graph n -> Graph (n+1)

Where G1 is the one-point graph with the point labeled $0$, and x :- g is g with a new vertex labeled $n$ attached, where x's binary expansion represents the newly added edges, where the indices correspond to the labels of the vertices. (on-bit indicates the vertex is connected by the edge)

But the implementations of graph structures I know in Haskell have fairly different data structures.

From Data.Graph from containers package:

type Graph = Array Vertex [Vertex]

From Data.Graph.Inductive.PatriciaTree from fgl package:

newtype Gr a b = Gr (GraphRep a b)

type GraphRep a b = IntMap (Context' a b)
type Context' a b = (IntMap [b], a, IntMap [b])

Where IntMap is from Data.IntMap.Lazy froim containers package.

So what structrue does the Graph Isomorphism Problem (especially in the paper by Babai & Helfgott) assume?