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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 ...
The construction is simple: place m vertices on the x-axis of the plane, avoiding the origin, with equal or nearly-equal numbers of points to the left and right of the y-axis. Similarly, place n vertices on the y -axis of the plane, avoiding the origin, with equal or nearly-equal numbers of points above and below the x -axis.
[2] [3] After a third chapter relating the crossing number to graph parameters including skewness, bisection width, thickness, and (via the Albertson conjecture) the chromatic number, the final chapter of part I concerns the computational complexity of finding minimum-crossing graph drawings, including the results that the problem is both NP ...
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 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 ...
Since each line segment lies on one of m lines, and any two lines intersect in at most one point, the crossing number of this graph is at most the number of points where two lines intersect, which is at most m(m − 1)/2. The crossing number inequality implies that either e ≤ 7.5n, or that m(m − 1)/2 ≥ e 3 / 33.75n 2.
For instance, the Szemerédi–Trotter theorem, an upper bound on the number of incidences that are possible between given numbers of points and lines in the plane, follows by constructing a graph whose vertices are the points and whose edges are the segments of lines between incident points. If there were more incidences than the Szemerédi ...