Extending simple monotone drawings

TL;DR

This work extends Levi’s Enlargement Lemma to arrangements of $x$-monotone pseudosegments, showing that any such planar arrangement is $(a,b)$-extendable by a simple $x$-monotone curve and that this leads to extending every simple monotone graph drawing to a complete graph. The authors develop a constructive, linear-time algorithm in the incidence size $m$ to produce the extending curve, using a careful lower-envelope inductive framework and a robust representation of the arrangement. They also explore cylindrically monotone and normal cylindrically monotone arrangements on the cylinder, proving extendability in the normal case via a planar reduction, while demonstrating obstructions and NP-hardness for the general cylindrically monotone setting. The results connect monotone drawing theory to Levi-like extension results, with practical implications for graph drawing and a clear boundary between tractable normal-cylinder cases and intractable general-cylinder cases.

Abstract

We prove the following variant of Levi's Enlargement Lemma: for an arbitrary arrangement $\mathcal{A}$ of $x$-monotone pseudosegments in the plane and a pair of points $a,b$ with distinct $x$-coordinates and not on the same pseudosegment, there exists a simple $x$-monotone curve with endpoints $a,b$ that intersects every curve of $\mathcal{A}$ at most once. As a consequence, every simple monotone drawing of a graph can be extended to a simple monotone drawing of a complete graph. We also show that extending an arrangement of cylindrically monotone pseudosegments is not always possible; in fact, the corresponding decision problem is NP-hard.

Figures (12)

• Figure 1: Left: a simple monotone drawing of a graph. Right: an extension of the drawing on the left to a simple monotone drawing of a complete graph. The added edges are dashed.
• Figure 2: An example of an arrangement of three pseudosegments that cannot be extended to pseudolines forming a pseudoline arrangement since any extension of the pseudosegments into pseudolines would contain a pair of pseudolines with two mutual intersections.
• Figure 3: Left: a simple cylindrically monotone drawing of $P_3+P_3$ where the edge $ab$ cannot be added Egg73_crossingKPRT15_saturatedK13_improved as a simple curve without crossing some edge of $P_3+P_3$. Since all edges of $P_3+P_3$ are incident to either $a$ or $b$, the added edge $ab$ would have at least two intersections with some edge of $P_3+P_3$. Right: a simple cylindrically monotone drawing of $P_3+P_2$ where the edge $ab$ cannot be added as a cylindrically monotone curve without intersecting some edge of $P_3+P_2$ twice.
• Figure 4: An arrangement of monotone pseudosegments with three added segments $\tau_1, \tau_2, \tau_3$ connecting points $a, b$ "from above".
• Figure 5: Induction step in the proof of Theorem \ref{['theorem_main']}. In the $i$th step (fourth step in the figure) we add pseudosegment $\gamma_i$ (dashed) intersecting the lower envelope (dotted) of the previous segments twice. The lower envelope remains a connected curve connecting $a$ with $b$ even after this addition.

Theorems & Definitions (8)

• Theorem 1
• Corollary 2
• Corollary 3
• Proposition 4
• Proposition 5
• Theorem 6
• Remark
• Lemma 7: AKPVSW23_inserting