A Reduction System for Optimal 1-Planar Graphs

TL;DR

This paper introduces a graph reduction system for optimal 1-planar graphs, enabling their reduction to irreducible extended wheel graphs while maintaining class properties, with efficient linear-time algorithms.

Contribution

It presents a context-sensitive reduction system for optimal 1-planar graphs that preserves graph class properties and operates efficiently in linear time.

Findings

Reductions can be computed in linear time.

Optimal 1-planar graphs can be reduced to extended wheel graphs within a size range.

The reduction system is non-deterministic and non-confluent.

Abstract

There is a graph reduction system so that every optimal 1-planar 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.