313 research outputs found
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
Disjoint Total Dominating Sets in Near-Triangulations
We show that every simple planar near-triangulation with minimum degree at
least three contains two disjoint total dominating sets. The class includes all
simple planar triangulations other than the triangle. This affirms a conjecture
of Goddard and Henning [Thoroughly dispersed colorings, J. Graph Theory, 88
(2018) 174-191]
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