A Reduction System for Optimal 1-Planar Graphs

There is a graph reduction system so that every optimal 1planar graph can be reduced to an irreducible extended wheel graph, provided the reductions are applied such that the given graph class is preserved. A graph is optimal 1-planar if it can be drawn in the plane with at most one crossing per edge and is optimal if it has the maximum of 4n− 8 edges. We show that the reduction system is context-sensitive so that the preservation of the graph class can be granted by local conditions which can be tested in constant time. Every optimal 1-planar graph G can be reduced to every extended wheel graph whose size is in a range from the (second) smallest one to some upper bound that depends on G. There is a reduction to the smallest extended wheel graph if G is not 5-connected, but not conversely. The reduction system has side effects and is non-deterministic and non-confluent. Nevertheless, reductions can be computed in linear time.