  S-prime graphs are graphs that cannot be represented as nontrivial subgraphs
of nontrivial Cartesian products of graphs, i.e., whenever it is a subgraph of
a nontrivial Cartesian product graph it is a subgraph of one the factors. A
graph is S-composite if it is not S-prime. Although linear time recognition
algorithms for determining whether a graph is prime or not with respect to the
Cartesian product are known, it remained unknown if a similar result holds also
for the recognition of S-prime and S-composite graphs.
  In this contribution the computational complexity of recognizing S-composite
and S-prime graphs is considered. Klav{\v{z}}ar \emph{et al.} [\emph{Discr.\
Math.} \textbf{244}: 223-230 (2002)] proved that a graph is S-composite if and
only if it admits a nontrivial path-$k$-coloring. The problem of determining
whether there exists a path-$k$-coloring for a given graph is shown to be
NP-complete even for $k=2$. This in turn is utilized to show that determining
whether a graph is S-composite is NP-complete and thus, determining whether a
graph is S-prime is CoNP-complete. Many other problems are shown to be NP-hard,
using the latter results.
