21 research outputs found
Cartesian product of hypergraphs: properties and algorithms
Cartesian products of graphs have been studied extensively since the 1960s.
They make it possible to decrease the algorithmic complexity of problems by
using the factorization of the product. Hypergraphs were introduced as a
generalization of graphs and the definition of Cartesian products extends
naturally to them. In this paper, we give new properties and algorithms
concerning coloring aspects of Cartesian products of hypergraphs. We also
extend a classical prime factorization algorithm initially designed for graphs
to connected conformal hypergraphs using 2-sections of hypergraphs
The Cartesian product of graphs with loops
We extend the definition of the Cartesian product to graphs with loops and
show that the Sabidussi-Vizing unique factorization theorem for connected
finite simple graphs still holds in this context for all connected finite
graphs with at least one unlooped vertex. We also prove that this factorization
can be computed in O(m) time, where m is the number of edges of the given
graph.Comment: 8 pages, 1 figur
Fast Recognition of Partial Star Products and Quasi Cartesian Products
This paper is concerned with the fast computation of a relation 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 is the relation , whose convex closure
yields the product relation that induces the prime factor
decomposition of connected graphs with respect to the Cartesian product. For
the construction of 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 for graphs with maximum bounded
degree.
Furthermore, we define \emph{quasi Cartesian products} as graphs with
non-trivial . 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
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