crossvertices

Crossvertices are vertices introduced at the intersection points of edges in a graph drawing, used to transform a nonplanar drawing into a planar representation for analysis or visualization.

Definition and construction

Given a graph G drawn in the plane with a set of edge crossings, the planarization of the drawing is obtained by placing a new vertex at each crossing and subdividing the involved edges there. In the resulting planar graph, each crossvertex represents a former crossing. If exactly two edges cross, a crossvertex typically has degree four; if more than two edges meet at the same point, the crossvertex degree increases accordingly. The operation is drawing-dependent: different embeddings of the same abstract graph can yield different sets of crossvertices.

Relation to planarity and analysis

Planarization yields a planar graph by construction, enabling the use of planar-graph algorithms on the drawn structure. The crossvertices encode the crossing information of the original drawing, so properties related to the embedding, such as the crossing number of that drawing, can be studied by counting crossvertices. The original abstract graph can be recovered in principle by removing the crossvertices and consolidating the edge fragments back into their original edges, though the process may depend on the drawing’s structure.

Applications

Crossvertices are used as a bookkeeping device in graph drawing and computational geometry. They facilitate planarity testing, embedding analysis, and the visualization of crossings. They also support simulations and heuristics that compare different drawings of the same graph, since the crossing information is made explicit as part of the graph’s structure.

Limitations

The concept is drawing-dependent; different representations of the same graph can yield different crossvertices and counts. Introducing crossvertices increases the graph’s size, which may affect the performance of algorithms. Crossvertices are a modeling tool rather than a fundamental property of the abstract graph.