On 4-Map Graphs and 1-Planar Graphs and their Recognition Problem

TL;DR

This paper explores the relationship between 1-planar graphs and 4-map graphs, providing polynomial-time recognition algorithms for certain subclasses of 1-planar graphs based on their structural properties.

Contribution

It establishes a one-to-one correspondence between 1-planar graphs and 4-map graphs, and develops polynomial-time recognition algorithms for specific subclasses of 1-planar graphs.

Findings

Recognition of crossing-augmented 1-planar graphs in degree 120 polynomial time

Recognition of fully triangulated 1-planar graphs in degree 3 polynomial time

Recognition of maximal 1-planar graphs in degree 5 polynomial time

Abstract

We establish a one-to-one correspondence between 1-planar graphs and general and hole-free 4-map graphs and show that 1-planar graphs can be recognized in polynomial time if they are crossing-augmented, fully triangulated, and maximal 1-planar, respectively, with a polynomial of degree 120, 3, and 5, respectively.