On the Complexity of Recognizing S-composite and S-prime Graphs

TL;DR

This paper investigates the computational complexity of recognizing S-prime and S-composite graphs, establishing that these recognition problems are NP-complete and CoNP-complete respectively, highlighting their computational difficulty.

Contribution

The paper proves that recognizing S-composite graphs is NP-complete and recognizing S-prime graphs is CoNP-complete, extending the understanding of their computational complexity.

Findings

Recognition of S-composite graphs is NP-complete.

Recognition of S-prime graphs is CoNP-complete.

Many related problems are NP-hard.

Abstract

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.