5 research outputs found
On the Complexity of Recognizing S-composite and S-prime Graphs
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--coloring. The problem of determining
whether there exists a path--coloring for a given graph is shown to be
NP-complete even for . 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
Strong Products of Hypergraphs: Unique Prime Factorization Theorems and Algorithms
It is well-known that all finite connected graphs have a unique prime factor
decomposition (PFD) with respect to the strong graph product which can be
computed in polynomial time. Essential for the PFD computation is the
construction of the so-called Cartesian skeleton of the graphs under
investigation.
In this contribution, we show that every connected thin hypergraph H has a
unique prime factorization with respect to the normal and strong (hypergraph)
product. Both products coincide with the usual strong graph product whenever H
is a graph. We introduce the notion of the Cartesian skeleton of hypergraphs as
a natural generalization of the Cartesian skeleton of graphs and prove that it
is uniquely defined for thin hypergraphs. Moreover, we show that the Cartesian
skeleton of hypergraphs can be determined in O(|E|^2) time and that the PFD can
be computed in O(|V|^2|E|) time, for hypergraphs H = (V,E) with bounded degree
and bounded rank