相关论文: Planar Linkages and Algebraic Sets

Three types of geometric structure---grid triangulations, rectangular subdivisions, and orthogonal polyhedra---can each be described combinatorially by a regular labeling: an assignment of colors and orientations to the edges of an…

We show that planar embeddable 3-connected CAD graphs are generically non-soluble. A CAD graph represents a configuration of points on the Euclidean plane with just enough distance dimensions between them to ensure rigidity. Formally, a CAD…

Entity-linking is a natural-language-processing task that consists in identifying the entities mentioned in a piece of text, linking each to an appropriate item in some knowledge base; when the knowledge base is Wikipedia, the problem comes…

This paper focuses on studying the configuration spaces of graphs realised in $\mathbb C^2$, such that the configuration space is, after normalisation, one dimensional. If this is the case, then the configuration space is, generically, a…

For a polygonal linkage, we produce a fast navigation algorithm on its configuration space. The basic idea is to approximate the configuration space by the vertex-edge graph of its cell decomposition discovered by the first author. The…

A segment representation of a graph is an assignment of line segments in 2D to the vertices in such a way that two segments intersect if and only if the corresponding vertices are adjacent. Not all graphs have such segment representations,…

A 1-planar graph is a graph which has a drawing on the plane such that each edge is crossed at most once. If a 1-planar graph is drawn in that way, the drawing is called a {\it 1-plane graph}. A graph is maximal 1-plane (or 1-planar) if no…

We show that any smooth one-dimensional link in the real projective three-plane is the fixed-point locus of a smooth symplectic surface in the complex projective three-plane which is invariant under complex conjugation. The degree of the…

Node-link diagrams are a popular method for representing graphs that capture relationships between individuals, businesses, proteins, and telecommunication endpoints. However, node-link diagrams may fail to convey insights regarding graph…

We introduce a notion of connected perimeter for planar sets defined as the lower semi-continuous envelope of perimeters of approximating sets which are measure-theoretically connected. A companion notion of simply connected perimeter is…

For a degree sequence, we define the set of edges that appear in every labeled realization of that sequence as forced, while the edges that appear in none as forbidden. We examine structure of graphs whose degree sequences contain either…

We consider the space of all configurations of finitely many (potentially nested) circles in the plane. We prove that this space is aspherical, and compute the fundamental group of each of its connected components. It turns out these…

The qualitative behavior of a dynamical system can be encoded in a graph. Each node of the graph is an equivalence class of chain-recurrent points and there is an edge from node $A$ to node $B$ if, using arbitrary small perturbations, a…

Given the degree sequence $d$ of a graph, the realization graph of $d$ is the graph having as its vertices the labeled realizations of $d$, with two vertices adjacent if one realization may be obtained from the other via an edge-switching…

In this paper we discuss the connected components of underlying graphs of halving lines' configurations. We show how to create a configuration whose underlying graph is the union of two given underlying graphs. We also prove that every…

This is an expository article on diagrammatic representations of knots and links in various settings via braids.

Flip graphs of non-crossing configurations in the plane are widely studied objects, e.g., flip graph of triangulations, spanning trees, Hamiltonian cycles, and perfect matchings. Typically, it is an easy exercise to prove connectivity of a…

An axis of a link projection is a closed curve which lies symmetrically on each region of the link projection. In this paper we define axis systems of link projections and characterize axis systems of the standard projections of twist…

A graph is near-planar if it can be obtained from a planar graph by adding an edge. We show the surprising fact that it is NP-hard to compute the crossing number of near-planar graphs. A graph is 1-planar if it has a drawing where every…

To any directed graph we associate an algebra with edges of the graph as generators and with relations defined by all pairs of directed paths with the same origin and terminus. Such algebras are related to factorizations of polynomials over…