Related papers: Covering half-grids with lines and planes
For a finite set $X$, we say that a set $H\subseteq X$ crosses a partition ${\cal P}=(X_1,\dots,X_k)$ of $X$ if $H$ intersects $\min (|H|,k)$ partition classes. If $|H|\geq k$, this means that $H$ meets all classes $X_i$, whilst for…
We study cross-graph charging schemes for graphs drawn in the plane. These are charging schemes where charge is moved across vertices of different graphs. Such methods have been recently applied to obtain various properties of…
The page number of a directed acyclic graph $G$ is the minimum $k$ for which there is a topological ordering of $G$ and a $k$-coloring of the edges such that no two edges of the same color cross, i.e., have alternating endpoints along the…
The Harary-Hill Conjecture states that for $n\geq 3$ every drawing of $K_n$ has at least \begin{align*} H(n) :=…
We establish new bounds on the number of tangencies and orthogonal intersections determined by an arrangement of curves. First, given a set of $n$ algebraic plane curves, we show that there are $O(n^{3/2})$ points where two or more curves…
Given a set $A$ of $n$ points (vertices) in general position in the plane, the \emph{complete geometric graph} $K_n[A]$ consists of all $\binom{n}{2}$ segments (edges) between the elements of $A$. It is known that the edge set of every…
We study the problem of covering a given point set in the plane by unit disks so that each point is covered exactly once. We prove that 17 points can always be exactly covered. On the other hand, we construct a set of 657 points where an…
A graph drawn on the plane is called $1$-plane if each edge is crossed at most once by another edge. In this paper, we show that every $4$-connected $1$-plane graph has a connected spanning plane subgraph. We also show that there exist…
In this paper, we study the many-to-many matching problem on planar point sets with integer coordinates: Given two disjoint sets $R,B \subset [\Delta]^2$ with $|R|+|B|=n$, the goal is to select a set of edges between $R$ and $B$ so that…
In this paper we investigate the structure of flip graphs on non-crossing perfect matchings in the plane. Specifically, consider all non-crossing straight-line perfect matchings on a set of $2n$ points that are placed equidistantly on the…
By definition, a rigid graph in $\mathbb{R}^d$ (or on a sphere) has a finite number of embeddings up to rigid motions for a given set of edge length constraints. These embeddings are related to the real solutions of an algebraic system.…
The concept of $k$-planarity is extensively studied in the context of Beyond Planarity. A graph is $k$-planar if it admits a drawing in the plane in which each edge is crossed at most $k$ times. The local crossing number of a graph is the…
Given a graph $G=(V,E)$ with costs on its edges, the minimum-cost edge cover problem consists of finding a subset of $E$ covering all vertices in $V$ at minimum cost. If $G$ is bipartite, this problem can be solved in time $O(|V|^3)$ via a…
We show that finding orthogonal grid-embeddings of plane graphs (planar with fixed combinatorial embedding) with the minimum number of bends in the so-called Kandinsky model (which allows vertices of degree $> 4$) is NP-complete, thus…
Let $S$ be a finite set of geometric objects partitioned into classes or \emph{colors}. A subset $S'\subseteq S$ is said to be \emph{balanced} if $S'$ contains the same amount of elements of $S$ from each of the colors. We study several…
Given a finite set of non-collinear points in the plane, there exists a line that passes through exactly two points. Such a line is called an ordinary line. An efficient algorithm for computing such a line was proposed by Mukhopadhyay et…
Suppose $A \in \mathbb{R}^{n \times n}$ is invertible and we are looking for the solution of $Ax = b$. Given an initial guess $x_1 \in \mathbb{R}$, we show that by reflecting through hyperplanes generated by the rows of $A$, we can generate…
Given a real algebraic curve, embedded in projective space, we study the computational problem of deciding whether there exists a hyperplane meeting the curve in real points only. More generally, given any divisor on such a curve, we may…
We revisit Hopcroft's problem and related fundamental problems about geometric range searching. Given $n$ points and $n$ lines in the plane, we show how to count the number of point-line incidence pairs or the number of point-above-line…
NOTE: Unfortunately, most of the results mentioned here were already known under the name of "d-separated interval piercing". The result that T_d(m) exists was first proved by Gya\'rfa\'s and Lehel in 1970, see [5]. Later, the result was…