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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 ...
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 ...
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
Each line produces three possibilities per point: the point can be in one of the two open half-planes on either side of the line, or it can be on the line. Two points can be considered to be equivalent if they have the same classification with respect to all of the lines.
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 ...
The line graphs of trees are exactly the claw-free block graphs. [23] These graphs have been used to solve a problem in extremal graph theory, of constructing a graph with a given number of edges and vertices whose largest tree induced as a subgraph is as small as possible. [24] All eigenvalues of the adjacency matrix A of a line graph are at ...