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 $\mathbb{R}^3$ and that any planar graph admits the same even in $\mathbb{R}^2$. For a graph $G$ and $d \in \{2,3\}$, let $\rho^1_d(G)$ denote the minimum number of lines in $\mathbb{R}^d$ that together can cover all edges of a drawing of $G$. For $d=2$, $G$ must be planar. We investigate the complexity of computing these parameters and obtain the following hardness and algorithmic results. - For $d\in\{2,3\}$, we prove that deciding whether $\rho^1_d(G)\le k$ for a given graph $G$ and integer $k$ is ${\exists\mathbb{R}}$-complete. - Since $\mathrm{NP}\subseteq{\exists\mathbb{R}}$, deciding $\rho^1_d(G)\le k$ is NP-hard for $d\in\{2,3\}$. On the positive side, we show that the problem is fixed-parameter tractable with respect to $k$. - Since ${\exists\mathbb{R}}\subseteq\mathrm{PSPACE}$, both $\rho^1_2(G)$ and $\rho^1_3(G)$ are computable in polynomial space. On the negative side, we show that drawings that are optimal with respect to $\rho^1_2$ or $\rho^1_3$ sometimes require irrational coordinates. - Let $\rho^2_3(G)$ be the minimum number of planes in $\mathbb{R}^3$ needed to cover a straight-line drawing of a graph $G$. We prove that deciding whether $\rho^2_3(G)\le k$ is NP-hard for any fixed $k \ge 2$. Hence, the problem is not fixed-parameter tractable with respect to $k$ unless $\mathrm{P}=\mathrm{NP}$

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..