As pointed out in:

http://www.ccs.neu.edu/research/demeter/design-decisions/products/

there is a large number of different graph products. There are even
more than shown in the Wolfram paper because we can take the product in terms
of nodes*nodes and edges*nodes. The traversal
graph uses a edges*nodes product.

In our paper we refer to automata theory. I think it is useful 
to describe the design space for products and to show that 
automata theory uses one element in the space and traversal graphs another.

For edges*nodes products:
For any two vertices:
(g, h) and (g',h')  in the product (first:edge, second:node), 
the adjacency of those two vertices is determined entirely by the adjacency 
(or equality, or non-adjacency) of g and g', and that of h and h'. 
There are  3*3-1 = 8 cases to be decided (three possibilities for each, 
with the case where both are equal eliminated) and thus there are  
256 = 2^8 different types of graph products that can be defined. 

For traversal graphs we use:
(g=g' (same strategy edge) and h adj h') or
(g adj g' with join point B and h=h'=B)

The second line gives an epsilon transition that is then transformed into
a regular transition.

The notion of automata product is used differently
than in AP. We make k copies of the class graph to
get a graph that has k * m nodes where k = O(n^2).
The traditional product has n*m nodes.

So the notion of traversal graph is inspired by automata products
or graph products. But it is not the same as the
standard automata product:
(g, h) and (g',h')  in the product (first:node, second:node),
(g adj g' with label a and  h adj h' with label a)


This is important to mention in future strategies papers.