Two-Page Book Embeddings of 4-Planar Graphs

TL;DR

This paper presents two algorithms for creating two-page book embeddings of 4-planar graphs, extending previous results for 3-planar graphs and utilizing hamiltonian cycle computations and book-embedding techniques.

Contribution

It introduces a linear-time algorithm for triconnected 4-planar graphs and a quadratic-time algorithm for general 4-planar graphs, advancing understanding of graph embeddings.

Findings

Linear-time algorithm for triconnected 4-planar graphs

Quadratic-time algorithm for general 4-planar graphs

Extension of subhamiltonian properties to 4-planar graphs

Abstract

Back in the Eighties, Heath showed that every 3-planar graph is subhamiltonian and asked whether this result can be extended to a class of graphs of degree greater than three. In this paper we affirmatively answer this question for the class of 4-planar graphs. Our contribution consists of two algorithms: The first one is limited to triconnected graphs, but runs in linear time and uses existing methods for computing hamiltonian cycles in planar graphs. The second one, which solves the general case of the problem, is a quadratic-time algorithm based on the book-embedding viewpoint of the problem.