Crosswences

Crosswences are a theoretical construct in graph theory used to describe a particular type of crossing in drawings of graphs. A crosswence occurs when two simple cycles cross each other at a point that is not a graph vertex, creating a local X-shaped intersection in the drawing. By definition, this crossing is transverse and cannot be removed by simply redrawing edges around existing vertices without changing the cycle structure of the graph.

Formal properties: Each crosswence involves four edge ends meeting at a non-vertex crossing point; it contributes one to the crossing number of the drawing. Crosswences are features of non-planar embeddings; removing them typically requires changing the embedding rather than altering local edge routes around existing vertices.

Examples: A minimal example is two triangles drawn so that their edges cross at an interior point. More generally, two cycles can cross transversely in multiple crosswences within a single drawing.

Applications: Crosswences serve as a conceptual tool in discussions of planarity obstructions and in developing graph-drawing algorithms that aim to minimize crossings. They also provide a simple analogy for teaching how different cycle structures constrain embeddings.

See also: crossing number; planar graph; non-planar graph; graph drawing.