The basis number of 1-planar graphs

2024-12-24

AI Summary

This paper investigates the cycle space basis number of 1-planar graphs—the minimum size of a *k*-bounded basis generating the Eulerian subspace, where each edge appears in at most *k* basis graphs. Method: Combining combinatorial graph theory, algebraic analysis of cycle spaces, embedding constructions, and extremal arguments, the study extends MacLane’s planarity criterion to non-planar embeddings and analyzes how crossing structures constrain basis numbers. Contribution/Results: It establishes the first systematic boundedness characterization: the basis number is unbounded over general 1-planar graphs, yet admits tight upper bounds—e.g., *b(G) ≤ 5*—for key subclasses including outerplanar, IC-planar, and 3-connected 1-planar graphs. Extremal constructions certify the tightness of these bounds. The work reveals a deep connection between basis number and crossing topology, offering a novel framework for studying topological invariants of non-planar graphs.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Graph-based Machine Learning
• Planning, Routing, and Scheduling: Other Foundations of Planning, Routing & Scheduling

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

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.

Problem

• 1-planar graphs
• cardinality problem
• finite condition

Innovation

• 1-planar graphs
• cardinality
• MacLane's criterion