Related papers: On Morphing 1-Planar Drawings
We consider graph Turing machines, a model of parallel computation on a graph, in which each vertex is only capable of performing one of a finite number of operations. This model of computation is a natural generalization of several…
We extend the concept of the law of a finite graph to graphings, which are, in general, infinite graphs whose vertices are equipped with the structure of a probability space. By doing this, we obtain a vast array of new unimodular measures.…
A connected graph can be associated with two distinct evolution algebras. In the first case, the structural matrix is the adjacency matrix of the graph itself. In the second case, the structural matrix is the transition probabilities matrix…
We show that the problem to decide whether two (convex) polytopes, given by their vertex-facet incidences, are combinatorially isomorphic is graph isomorphism complete, even for simple or simplicial polytopes. On the other hand, we give a…
We present a simplified exposition of some classical and modern results on graph drawings in the plane. These results are chosen so that they illustrate some spectacular recent higher-dimensional results on the border of topology and…
We study invariants of virtual graphoids, which are virtual spatial graph diagrams with two distinguished degree-one vertices modulo graph Reidemeister moves applied away from the distinguished vertices. Generalizing previously known…
Given the implicit equation $F(x,y,t,s)$ of a family of algebraic plane curves depending on the parameters $t,s$, we provide an algorithm for studying the topology types arising in the family. For this purpose, the algorithm computes a…
In this paper we present algorithms for computing the topology of planar and space rational curves defined by a parametrization. The algorithms given here work directly with the parametrization of the curve, and do not require to compute or…
Fractional graph isomorphism is the linear relaxation of an integer programming formulation of graph isomorphism. It preserves some invariants of graphs, like degree sequences and equitable partitions, but it does not preserve others like…
Simple drawings are drawings of graphs in which any two edges intersect at most once (either at a common endpoint or a proper crossing), and no edge intersects itself. We analyze several characteristics of simple drawings of complete…
As two fundamental problems, graph cuts and graph matching have been investigated over decades, resulting in vast literature in these two topics respectively. However the way of jointly applying and solving graph cuts and matching receives…
Graphs are used in many disciplines to model the relationships that exist between objects in a complex discrete system. Researchers may wish to compare a network of interest to a "typical" graph from a family (or ensemble) of graphs which…
In this paper, we introduce the notion of a dual topological graph of a given topological graph, and show that it defines a C*-algebra isomorphic to the C*-algebra of the given one. Repeating to take a dual, and taking a projective limit,…
During recent years, the renaissance of neural networks as the major machine learning paradigm and more specifically, the confirmation that deep learning techniques provide state-of-the-art results for most of computer vision tasks has been…
Homeomorphisms allowing us to prove topological equivalences between one-parameter families of maps undergoing the same bifurcation are constructed in this paper. This provides a solution for a classical problem in bifurcation theory that…
A graph is beyond-planar if it can be drawn in the plane with a specific restriction on crossings. Several types of beyond-planar graphs have been investigated, such as k-planar if every edge is crossed at most k times and RAC if edges can…
A graph is $1$-planar if it has a drawing in the plane such that each edge is crossed at most once by another edge. Moreover, if this drawing has the additional property that for each crossing of two edges the end vertices of these edges…
We study the topology associated with physical vector and scalar fields. A mathematical object, e.g., a ball, can be continuously deformed, without tearing or gluing, to make other topologically equivalent objects, e.g., a cube or a solid…
We prove that for any weakly convergent sequence of finite graphs with bounded vertex degrees, there exists a topological limit graphing.
Various simplicial complexes can be associated with a graph. Box complexes form an important families of such simplicial complexes and are especially useful for providing lower bounds on the chromatic number of the graph via some of their…