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In graph theory, the crossing number cr(G) of a graph G is the lowest number of edge crossings of a plane drawing of the graph G. For instance, a graph is planar if and only if its crossing number is zero. Determining the crossing number continues to be of great importance in graph drawing, as user studies have shown that drawing graphs with ...
Crossing Numbers of Graphs is a book in mathematics, on the minimum number of edge crossings needed in graph drawings. It was written by Marcus Schaefer, a professor of computer science at DePaul University , and published in 2018 by the CRC Press in their book series Discrete Mathematics and its Applications.
It states that, for graphs where the number e of edges is sufficiently larger than the number n of vertices, the crossing number is at least proportional to e 3 /n 2. It has applications in VLSI design and combinatorial geometry, and was discovered independently by Ajtai, Chvátal, Newborn, and Szemerédi [1] and by Leighton. [2]
Planarization, a planar graph formed from a drawing with crossings by replacing each crossing point by a new vertex; Thickness (graph theory), the smallest number of planar graphs into which the edges of a given graph may be partitioned; Planarity, a puzzle computer game in which the objective is to embed a planar graph onto a plane
A crossing is counted whenever two edges that are disjoint in the graph have a nonempty intersection in the plane. The question is then, what is the minimum number of crossings in such a drawing? [2] [3] Turán's formulation of this problem is often recognized as one of the first studies of the crossing numbers of graphs. [4]
A graph with odd-crossing number 13 and pair-crossing number 15 [1]. In mathematics, a topological graph is a representation of a graph in the plane, where the vertices of the graph are represented by distinct points and the edges by Jordan arcs (connected pieces of Jordan curves) joining the corresponding pairs of points.
Graph drawing also can be said to encompass problems that deal with the crossing number and its various generalizations. The crossing number of a graph is the minimum number of intersections between edges that a drawing of the graph in the plane must contain. For a planar graph, the crossing number is zero by definition. Drawings on surfaces ...
Every 1-planar graph with n vertices has at most 4n − 8 edges. [4] More strongly, each 1-planar drawing has at most n − 2 crossings; removing one edge from each crossing pair of edges leaves a planar graph, which can have at most 3n − 6 edges, from which the 4n − 8 bound on the number of edges in the original 1-planar graph immediately follows. [5]