The reducibility of optimal 1-planar graphs

TL;DR

This paper characterizes when optimal 1-planar graphs can be decomposed into smaller graphs via lexicographic product, addressing a problem related to 1-planarity of certain graph products.

Contribution

It provides a characterization of reducibility for optimal 1-planar graphs, advancing understanding of their structural properties.

Findings

Identifies conditions under which optimal 1-planar graphs are reducible.

Addresses a 2015 open problem on 1-planarity of lexicographic products.

Enhances the theoretical framework of 1-planar graph structure.

Abstract

A graph is reducible if it is the lexicographic product of two smaller non-trivial graphs. It is well-known a 1-planar graph with $n ~(\ge3)$ vertices has at most $4n-8$ edges, and a graph $G$ with $n$ vertices is optimal if $G$ has exactly $4n-8$ edges. In this paper, we characterize the reducibility of optimal 1-planar graphs. This work is motivated by a problem posed by Bucko and Czap in 2015, which concerns determining the 1-planarity of the lexicographic product of a graph and two isolated vertices.