Edge densities of drawings of graphs with one forbidden cell

TL;DR

This work initiates the systematic study of graph drawings that avoid a fixed cell type, examining how forbidding a single cell pattern shapes edge density across drawing styles (arbitrary, simple, non-homotopic) and graph classes (multigraphs, simple graphs). The authors adapt discharging arguments and construct explicit examples to establish upper and lower bounds for each cell type, with most results showing linear or superlinear density and many bounds tight up to additive constants; they also classify when simple drawings can avoid cell types not incident to crossings and improve the known lower bound for non-homotopic quasiplanar drawings to $7.5n-28$. Moreover, the paper reveals separations between quasiplanar and $\mathfrak{c}$-free drawings, provides quadratic-density constructions on the torus for certain cell types, and highlights several open problems, including the precise density of simple $\mathfrak{c}$-free plane drawings. Overall, the results deepen understanding of density questions in beyond-planar graph classes and introduce a versatile framework based on cell-type avoidance and discharging techniques.

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 \celltype, 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 \celltype that the edge density of $n$-vertex (multi)graphs with \celltype-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 cell types that are not incident to any crossing in more detail and find that all connected simple graphs but a short list of exceptions admit a simple drawing that does not contain any such cells. 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 $7.5n-28$.

Figures (6)

• Figure 1: A simple drawing of $K_n$ (here $n=8$) containing only cells of size at most $5$, and one cell of size $2n$.
• Figure 2: A simple -free drawing of $K_{12}$. (A small perturbation eliminates the multi-crossings.) Edges of length three are drawn in blue and those of length two in red. Edges of length $n/2$ are depicted in green and only partially drawn.
• Figure 3: A crossing can be redrawn to bound only cells of size 2 (shaded gray) or at least $6$.
• Figure 5: Left: Removing a vertex $v$ and its incident edge segments, a new cell $c_v$ (shaded gray) is formed. Others: Case distinction in \ref{['thm:threezero_ub']} where $c_v$ is a -cell.
• Figure 8: Illustration of \ref{['con:non-quasi-3_0-free']}. By adding a vertex and an edge, the number of -cells in a drawing can be reduced by one.

Theorems & Definitions (8)

• Lemma 2
• Lemma 3
• Theorem 4
• Theorem 5
• Proposition 6
• Proposition 7
• Corollary 8
• Conjecture 9