Perfect weak modular product graphs

In this paper we enumerate the necessary and sufficient conditions for the weak modular product of two simple graphs to be perfect. The weak modular product differs from the direct product by also encoding non-adjacencies of the factor graphs in its edges. This work is motivated by the following: a 1978 theorem of Kozen states that two graphs on $n$ vertices are isomorphic if and only if there is a clique of size $n$ in the weak modular product between the two graphs. Furthermore, a straightforward corollary of Kozen's theorem and Lov\'{a}sz's sandwich theorem is if the weak modular product between two graphs is perfect, then checking if the graphs are isomorphic is polynomial in $n$. Interesting cases include complete multipartite graphs and disjoint unions of cliques. All perfect weak modular products have factors that fall into classes of graphs for which testing isomorphism is already known to be polynomial in the number of vertices.