Related papers: Enumerating topological $(n_k)$-configurations
An $(n_k)$ configuration is a set of $n$ points and $n$ lines such that each point lies on $k$ lines while each line contains $k$ points. The configuration is geometric, topological, or combinatorial depending on whether lines are…
This paper begins by extending the notion of a combinatorial configuration of points and lines to a combinatorial configuration of points and planes that we refer to as configurations of order $2$. We then proceed to investigate a further…
A $\Bbbk$-configuration is a set of points $\mathbb{X}$ in $\mathbb{P}^2$ that satisfies a number of geometric conditions. Associated to a $\Bbbk$-configuration is a sequence $(d_1,\ldots,d_s)$ of positive integers, called its type, which…
A generic method for combinatorial constructions of intrinsic geometrical spaces is presented. It is based on the well known inverse sequences of finite graphs that determine (in the limit) topological spaces. If a pattern of the…
In a series of papers and in his 2009 book on configurations Branko Gr\"unbaum described a sequence of operations to produce new $(n_{4})$ configurations from various input configurations. These operations were later called the "Gr\"unbaum…
We start by introducing the basics of configurations of points and lines, and then move into discussing symmetry groups of these configurations. Specifically, we explore how we might classify the symmetries of $(9_3)$ and $(10_3)$ geometric…
We characterize the topological configurations of points and lines that may arise when placing n points on a circle and drawing the n perpendicular bisectors of the sides of the corresponding convex cyclic n-gon. We also provide exact and…
Topological drawings are natural representations of graphs in the plane, where vertices are represented by points, and edges by curves connecting the points. Topological drawings of complete graphs and of complete bipartite graphs have been…
We describe a new way to construct finite geometric objects. For every k we obtain a symmetric configuration E(k-1) with k points on a line. In particular, we have a constructive existence proof for such configurations. The method is very…
We study combinatorial configurations with the associated point and line graphs being strongly regular. Examples not belonging to known classes such as partial geometries and their generalizations or elliptic semiplanes are constructed.…
A simple topological graph $G$ is a graph drawn in the plane so that any pair of edges have at most one point in common, which is either an endpoint or a proper crossing. $G$ is called saturated if no further edge can be added without…
A (v,b,r,k) combinatorial configuration is a (r,k)-biregular bipartite graph with v vertices on the left and b vertices on the right and with no cycle of length 4. Combinatorial configurations have become very important for some…
We study the smallest possible number of points in a topological space having k open sets. Equivalently, this is the smallest possible number of elements in a poset having k order ideals. Using efficient algorithms for constructing a…
It is well known that not every combinatorial configuration admits a geometric realization with points and lines. Moreover, some of them do not even admit realizations with pseudoline arrangements, i.e., they are not topological. In this…
We present some methods for constructing connected spatial geometric configurations $(p_{q}, n_{k})$ of points and lines, preserved by the same rotations (and reflections) of Euclidean space $E^{3}$ as the chosen Platonic solid. In this…
In many areas of applied geometric/numeric computational mathematics, including geo-mapping, computer vision, computer graphics, finite element analysis, medical imaging, geometric design, and solid modeling, one has to compute incidences,…
If $G$ is a graph with vertex set $V$, let Conf$_n^{\text{sink}}(G,V)$ be the space of $n$-tuples of points on $G$, which are only allowed to overlap on elements of $V$. We think of Conf$_n^{\text{sink}}(G,V)$ as a configuration space of…
Computational topology is an area that revisits topological problems from an algorithmic point of view, and develops topological tools for improved algorithms. We survey results in computational topology that are concerned with graphs drawn…
A clustering algorithm partitions a set of data points into smaller sets (clusters) such that each subset is more tightly packed than the whole. Many approaches to clustering translate the vector data into a graph with edges reflecting a…
The number of topologies and non-homeomorphic topologies on a fixed finite set are now known up to $n=18$, $n=16$ but still no complete formula yet (Sloane). There are one to one correspondence among topologies, preorder and digraphs. In…