Related papers: Computing the Geometric Intersection Number of Cur…
The geometry of a graph $G$ embedded on a closed oriented surface $S$ can be probed by counting the intersections of $G$ with closed curves on $S$. Of special interest is the map $c \mapsto \mu_G(c)$ counting the minimum number of…
We give a recipe to compute the geometric intersection number of an integral lamination with a particular type of integral lamination on an n-times punctured disk. This provides a way to find the geometric intersection number of two…
We consider systems of simple closed curves on surfaces and their total number of intersection points, their so-called crossing number. For a fixed number of curves, we aim to minimise the crossing number. We determine the minimal crossing…
We describe a new algorithm to compute the geometric intersection number between two curves, given as edge vectors on an ideal triangulation. Most importantly, this algorithm runs in polynomial time in the bit-size of the two edge vectors.…
We present some algorithms that provide useful topological information about curves in surfaces. One of the main algorithms computes the geometric intersection number of two properly embedded 1-manifolds $C_1$ and $C_2$ in a compact…
A geometric graph is a graph whose vertices are points in general position in the plane and its edges are straight line segments joining these points. In this paper we give an $O(n^2 \log n)$ algorithm to compute the number of pairs of…
We initiate the study of computing shortest non-separating simple closed curves with some given topological properties on non-orientable surfaces. While, for orientable surfaces, any two non-separating simple closed curves are related by a…
In an orientable surface with boundary, free homotopy classes of curves on surfaces are in one to one correspondence with cyclic reduced words in a set of standard generators of the fundamental group. The combinatorial length of a class is…
This paper explores the relationship between closed curves on surfaces and their intersections. Like Dehn-Thurston coordinates for simple curves, we explore how to determine closed curves using the number of times they intersect other…
Given an ordered sequence of $N$-choose-2 integers, we give necessary and sufficient conditions to have an ordered collection of $N$ simple closed curves on a torus such that the algebraic pairwise intersections of those curves are the…
We initiate the study of diameter computation in geometric intersection graphs from the fine-grained complexity perspective. A geometric intersection graph is a graph whose vertices correspond to some shapes in $d$-dimensional Euclidean…
Based on Nielsen fixed point theory and Gr\"{o}bner-Shirshov basis, we obtain a simple method to compute geometric intersection numbers and self-intersection geometric numbers of loops on surfaces.
A polygon C is an intersecting polygon for a set O of objects in the plane if C intersects each object in O, where the polygon includes its interior. We study the problem of computing the minimum-perimeter intersecting polygon and the…
Computing the diameter of the intersection graphs of objects is a basic problem in computational geometry. Previous works showed that the complexity of computing the diameter mainly depends on the object types: for unit disks and squares in…
We consider the problem of cutting a set of edges on a polyhedral manifold surface, possibly with boundary, to obtain a single topological disk, minimizing either the total number of cut edges or their total length. We show that this…
We study mapping class group orbits of homotopy and isotopy classes of curves with self-intersections. We exhibit the asymptotics of the number of such orbits of curves with a bounded number of self-intersections, as the complexity of the…
We study the following separation problem: Given a collection of colored objects in the plane, compute a shortest "fence" $F$, i.e., a union of curves of minimum total length, that separates every two objects of different colors. Two…
On an orientable surface $S$, consider a collection $\Gamma$ of closed curves. The (geometric) intersection number $i_S(\Gamma)$ is the minimum number of self-intersections that a collection $\Gamma'$ can have, where $\Gamma'$ results from…
For two oriented simple closed curves on a compact orientable surface with a connected boundary we introduce a simple computation of a value in the first homology group of the surface, which detects in some cases that the geometric…
Oriented closed curves on an orientable surface with boundary are described up to continuous deformation by reduced cyclic words in the generators of the fundamental group and their inverses. By self-intersection number one means the…