9,464 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
A combinatorial approach to knot recognition
This is a report on our ongoing research on a combinatorial approach to knot
recognition, using coloring of knots by certain algebraic objects called
quandles. The aim of the paper is to summarize the mathematical theory of knot
coloring in a compact, accessible manner, and to show how to use it for
computational purposes. In particular, we address how to determine colorability
of a knot, and propose to use SAT solving to search for colorings. The
computational complexity of the problem, both in theory and in our
implementation, is discussed. In the last part, we explain how coloring can be
utilized in knot recognition
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
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