  This paper is concerned with the fast computation of a relation $\R$ on the
edge set of connected graphs that plays a decisive role in the recognition of
approximate Cartesian products, the weak reconstruction of Cartesian products,
and the recognition of Cartesian graph bundles with a triangle free basis.
  A special case of $\R$ is the relation $\delta^\ast$, whose convex closure
yields the product relation $\sigma$ that induces the prime factor
decomposition of connected graphs with respect to the Cartesian product. For
the construction of $\R$ so-called Partial Star Products are of particular
interest. Several special data structures are used that allow to compute
Partial Star Products in constant time. These computations are tuned to the
recognition of approximate graph products, but also lead to a linear time
algorithm for the computation of $\delta^\ast$ for graphs with maximum bounded
degree.
  Furthermore, we define \emph{quasi Cartesian products} as graphs with
non-trivial $\delta^\ast$. We provide several examples, and show that quasi
Cartesian products can be recognized in linear time for graphs with bounded
maximum degree. Finally, we note that quasi products can be recognized in
sublinear time with a parallelized algorithm.
