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This is the minimum number of crossings among all drawings of this graph, so the graph has crossing number cr(G) = 3. 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 ...
A simple proof of this follows from the crossing number inequality: [15] if cells have a total of + edges, one can form a graph with nodes (one per cell) and edges (one per pair of consecutive cells on the same line). The edges of this graph can be drawn as curves that do not cross within the cells corresponding to their endpoints, and then ...
The slope number of a graph is the minimum number of distinct edge slopes needed in a drawing with straight line segment edges (allowing crossings). Cubic graphs have slope number at most four, but graphs of degree five may have unbounded slope number; it remains open whether the slope number of degree-4 graphs is bounded. [12]
A convex curve (black) forms a connected subset of the boundary of a convex set (blue), and has a supporting line (red) through each of its points. A parabola, a convex curve that is the graph of the convex function () = In geometry, a convex curve is a plane curve that has a supporting line through each of its points.
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]
[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 ...
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]
By using homogeneous coordinates, the intersection point of two implicitly defined lines can be determined quite easily. In 2D, every point can be defined as a projection of a 3D point, given as the ordered triple (x, y, w). The mapping from 3D to 2D coordinates is (x′, y′) = ( x / w , y / w ).