The basis number of 1-planar graphs

TL;DR

This paper investigates the basis number of 1-planar graphs, showing it is unbounded in general but bounded for many subclasses, extending the understanding of cycle space generation in near-planar graphs.

Contribution

It introduces the concept of basis number for 1-planar graphs and establishes bounds, expanding cycle space theory beyond planar graphs.

Findings

Basis number is unbounded for general 1-planar graphs.

Many subclasses of 1-planar graphs have bounded basis numbers.

Extension of MacLane's planarity criterion to 1-planar graphs.

Abstract

Let $B$ be a set of Eulerian subgraphs of a graph $G$. We say $B$ forms a $k$-basis if it is a minimum set that generates the cycle space of $G$, and any edge of $G$ lies in at most $k$ members of $B$. The basis number of a graph $G$, denoted by $b(G)$, is the smallest integer such that $G$ has a $k$-basis. A graph is called 1-planar (resp. planar) if it can be embedded in the plane with at most one crossing (resp. no crossing) per edge. MacLane's planarity criterion characterizes planar graphs based on their cycle space, stating that a graph is planar if and only if it has a $2$-basis. We study here the basis number of 1-planar graphs, demonstrate that it is unbounded in general, and show that it is bounded for many subclasses of 1-planar graphs.