Edge densities of drawings of graphs with one forbidden cell

TL;DR

This paper systematically studies the maximum edge density of graphs with drawings that exclude a specific cell type, providing bounds and characterizations across various drawing styles and graph classes.

Contribution

It introduces the concept of -free drawings, analyzes edge density bounds for all cell types, and characterizes simple graphs avoiding certain cell types.

Findings

Edge density is linear or superlinear for most cell types.

Bounds are tight up to an additive constant in many cases.

Improved lower bound on edge density for non-homotopic quasiplanar drawings.

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 triangular cell, Ackerman and Tardos [JCTA 2007] proved that $G$ has at most $8n-20$ edges, while Kaufmann, Klemz, Knorr, Reddy, Schr\"oder, and Ueckerdt [GD 2024] showed that this bound is tight. In this paper, we initiate the in-depth study of $\mathfrak{c}$-free drawings, that is, drawings that do not contain any cell of one fixed cell type $\mathfrak{c}$, and investigate the edge density of the corresponding graphs, 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 cell type $\mathfrak{c}$. 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 linear in $n$ or superlinear in $n$. In most cases, our bounds are tight up to an additive constant. We further consider the question which simple graphs admit a simple drawing without some given cell type(s). For the class of cell types that are not incident to any crossing, we give a complete characterization of all simple graphs that admit a simple drawing without any such cell. 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 $7.5n-28$.