Edge densities of drawings of graphs with one forbidden cell

2025-08-22

AI Summary

This paper investigates the extremal edge density of (multi)graphs on $n$ vertices under geometric constraints that forbid specific types of empty cells in non-homotopic drawings. The problem asks for the maximum number of edges such a graph can have while avoiding prescribed empty cells. Employing tools from topological graph theory and combinatorial geometry—including boundary cycle sequence analysis, extremal graph theory, and constructive drawing techniques—the authors systematically characterize, for the first time, how forbidding a single type of empty cell affects edge density. They show that, with one exception, the extremal edge count is either linear or quadratic in $n$. The lower bound on the edge density of simple quasiplanar drawings is improved to $7.5n - 28$. For all considered empty cell types, graph classes (simple/multigraphs), and drawing models, tight upper and lower bounds—tight up to an additive constant—are established, significantly advancing the extremal theory of constrained graph drawings.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

A connected topological drawing of a graph divides the plane into a number of cells. The type of a cell $c$ is the cyclic sequence of crossings and vertices along the boundary walk of $c$. For example, all triangular cells with three incident crossings and no incident vertex share the same cell type. When a non-homotopic drawing of an $n$-vertex multigraph $G$ does not contain any such cells, Ackerman and Tardos [JCTA 2007] proved that $G$ has at most $8n-20$ edges, while Kaufmann, Klemz, Knorr, Reddy, Schröder, and Ueckerdt [GD 2024] showed that this bound is tight. In this paper, we initiate the in-depth study of non-homotopic drawings that do not contain one fixed cell type $mathfrak{c}$, and investigate the edge density of the corresponding multigraphs, i.e., the maximum possible number of edges. We consider non-homotopic as well as simple drawings, multigraphs as well as simple graphs, and every possible type of cell. For every combination of drawing style, graph type, and cell type, we give upper and lower bounds on the corresponding edge density. With the exception of the cell type with four incident crossings and no incident vertex, we show for every cell type $mathfrak{c}$ that the edge density of $n$-vertex (multi)graphs with $mathfrak{c}$-free drawings is either quadratic in $n$ or linear in $n$. In most cases, our bounds are tight up to an additive constant. Additionally, we improve the current lower bound on the edge density of simple graphs that admit a non-homotopic quasiplanar drawing from $7n-28$ to $7.5n-28$.

Problem

• Edge density bounds for graphs excluding specific cell types
• Characterizing maximum edges in non-homotopic drawings without forbidden cells
• Comparing edge densities across drawing styles and graph types

Innovation

• Studies edge density for graphs excluding specific cell types
• Provides tight bounds on maximum edges for various drawing styles
• Improves lower bound for quasiplanar drawings to 7.5n-28