Recognizing Optimal 1-Planar Graphs in Linear Time

TL;DR

This paper presents a linear-time algorithm for recognizing optimal 1-planar graphs by reducing them to irreducible extended wheel graphs using a non-deterministic reduction system.

Contribution

It introduces a novel linear-time recognition algorithm for optimal 1-planar graphs based on a graph reduction system.

Findings

Recognition of optimal 1-planar graphs is achievable in linear time.

The reduction system effectively simplifies graphs to irreducible extended wheel graphs.

The algorithm operates with a non-deterministic, constraint-based, non-confluent reduction process.

Abstract

A graph with n vertices is 1-planar if it can be drawn in the plane such that each edge is crossed at most once, and is optimal if it has the maximum of 4n-8 edges. We show that optimal 1-planar graphs can be recognized in linear time. Our algorithm implements a graph reduction system with two rules, which can be used to reduce every optimal 1-planar graph to an irreducible extended wheel graph. The graph reduction system is non-deterministic, constraint, and non-confluent.