The basis number of 1-planar graphs

TL;DR

This work studies the basis number $b(G)$ of $1$-planar graphs by examining the cycle space $\mathcal{C}(G)$ and $k$-bases. It shows that $b(G)$ can be unbounded overall for $1$-planar graphs, even under degree constraints or certain crossing restrictions, but identifies substantial subclasses with bounded basis numbers, notably when the skeleton $\mathsf{sk}(G)$ is connected (yielding $b(G)\le 4$) and, with further structural restrictions, $b(G)\le 3$. The authors develop a toolkit of graph-operations (contraction, edge addition, subdivision, edge-replacement) and decomposition techniques (two-subgraph covers, auxiliary graphs) that preserve or bound the basis number, and they introduce concepts specific to $1$-planarity such as poppy and full-crossing drawings to derive bounds. The paper also provides concrete examples and open questions, including the minimal size of a $1$-planar graph with $b(G)\ge 4$ and the complexity of determining $b(G)$ for $1$-planar graphs, highlighting both the limits of current methods and directions for future work.

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.

Figures (17)

• Figure 1: A $1$-plane graph with a 3-basis.
• Figure 2: Replacing edge $e$ of $G$ by a 2-connected planar graph $H$ with terminals $s,t$. The paths in $\mathcal{P}$ are red (dotted) and blue (dashed).
• Figure 3: The Franklin graph and its skeleton in a 1-planar drawing (taken from cubic1-planarity).
• Figure 4: The skirts of $x$ are red (dotted), orange (short dotted), blue (dot-dot-dashed), and green (dashed).
• Figure 5: (a) Crossing $x$ defines four skirt-walks which together with $x$ form a poppy. (b) Petersen graph. (c) Petersen graph as a poppy 1-planar graph.

Theorems & Definitions (49)

• Lemma 1.1: MacLane’s planarity criterion maclane1970combinatorial
• Lemma 2.1
• Definition 2.2
• Example 2.3
• Lemma 3.1
• proof
• Corollary 3.2
• Proposition 3.3
• proof
• Corollary 3.4