The density of maximal IC-plane graphs and maximal NIC-plane graphs

TL;DR

This work determines the exact minimal edge density of maximal IC-plane and NIC-plane graphs on n vertices. By analyzing crossing structures, face types, and the role of hermits, the authors reduce to a pivotal special-case configuration and derive tight lower bounds: $|E|\ge\left\lceil\frac{7}{3}n-\frac{14}{3}\right\rceil$ for IC-plane graphs and $|E|\ge\left\lceil\frac{11}{5}n-\frac{18}{5}\right\rceil$ for NIC-plane graphs, with constructions showing tightness for infinitely many n. The general case is then established via targeted augmentations that transform arbitrary maximal IC-/NIC-plane graphs into the special-case form, confirming the bounds for all n. These results resolve a longstanding question about the density of saturated drawings in these graph classes and provide exact, verifiable formulas for their minimal edge counts.

Abstract

In this paper, we show that any maximal IC-plane graph of order $n$ has at least $\left\lceil\frac{7}{3}n-\frac{14}{3}\right\rceil$ edges, and any maximal NIC-plane graph of order $n$ has at least $\left\lceil\frac{11}{5}n-\frac{18}{5}\right\rceil$ edges. Moreover, we show that both results are tight for infinitely many integers $n$.

Figures (9)

• Figure 1: Two different IC-planar drawings of an IC-planar graph $G$. The drawing on the left is saturated, while drawing on the right is not. Hence, $G$ is not a maximal IC-planar graph.
• Figure 2: Possible types of faces in $G$.
• Figure 3: The region $R_{5}$ on the left is infinite, and on the right is finite.
• Figure 4: $H^{*}$, $H_{1}$ and $H_{2}$.
• Figure 5: $M^{*}$, $M_{1}$ and $M_{2}$.

Theorems & Definitions (14)

• Theorem 1
• Theorem 2
• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Proposition 1
• Proposition 2
• Proposition 3
• Claim 1