6 research outputs found
The Complexity of Drawing Graphs on Few Lines and Few Planes
It is well known that any graph admits a crossing-free straight-line drawing
in and that any planar graph admits the same even in
. For a graph and , let denote
the minimum number of lines in that together can cover all edges
of a drawing of . For , must be planar. We investigate the
complexity of computing these parameters and obtain the following hardness and
algorithmic results.
- For , we prove that deciding whether for a
given graph and integer is -complete.
- Since , deciding is NP-hard for . On the positive side, we show that the problem
is fixed-parameter tractable with respect to .
- Since , both and
are computable in polynomial space. On the negative side, we show
that drawings that are optimal with respect to or
sometimes require irrational coordinates.
- Let be the minimum number of planes in needed
to cover a straight-line drawing of a graph . We prove that deciding whether
is NP-hard for any fixed . Hence, the problem is
not fixed-parameter tractable with respect to unless
Загадки Велесової книги
Дана публікація розкриває помилки попередніх досліджень «Велесової книги» та надає пояснення важкодоступних висловів тексту.Данная публикация раскрывает ошибки предыдущих исследований «Велесовой книги» и дает объяснение труднодоступных выражений в тексте.This publication reveals the mistakes of the former researches on the «Veles-book» and gives the meanings of some hard-to-understand terms of the text
A Dynamic Setup for Elementary Geometry
In this article we survey the theoretical background that is required to build a consistent and continuous setup of dynamic elementary geometry. Unlike i
Some Algorithmic Problems in Polytope Theory
Convex polytopes, i.e.. the intersections of finitely many closed affine halfspaces in R^d, are important objects in various areas of mathematics and other disciplines. In particular, the compact..