On subgraphs of Cartesian product graphs
Discrete Mathematics - Algebraic and topological methods in graph theory
The complexity of satisfiability problems
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
Characterizing subgraphs of Hamming graphs
Journal of Graph Theory
Topics in Graph Theory: Graphs and Their Cartesian Product
Topics in Graph Theory: Graphs and Their Cartesian Product
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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. Klavzar et al. [S. Klavzar, A. Lipovec, M. Petkovsek, On subgraphs of Cartesian product graphs. Discrete Math., 244 (2002) 223-230] 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. A plenty of other problems are shown to be NP-hard, using the latter results.