On 1-bend Upward Point-set Embeddings of $st$-digraphs

TL;DR

This work addresses upward point-set embeddings with a single bend per edge on one-sided convex point sets for $st$-digraphs. It shows a positive result: every $st$-outerplanar graph admits a $1$-bend UPSE on any upward-one-sided convex (UOSC) point set, via HP-completion and a careful 2-page upward book-embedding framework with slope-guided drawings. It also provides a sharp negative result: for every $n \ge 18$ there exists a $2$-outerplanar $st$-digraph with $n$ vertices and a UOSC point set $S$ for which no $1$-bend UPSE exists on $S$, highlighting structural obstructions. The approach combines a decomposition into cores and appendages, forbidden configurations to characterize obstructions, and a slope-assignment technique to realize embeddings with at most one bend per edge. Overall, the paper delineates the boundary between tractable and intractable instances for $1$-bend UPSE on UOSC point sets in the $st$-digraph setting, with implications for graph drawing on constrained point sets.

Abstract

We study the upward point-set embeddability of digraphs on one-sided convex point sets with at most 1 bend per edge. We provide an algorithm to compute a 1-bend upward point-set embedding of outerplanar $st$-digraphs on arbitrary one-sided convex point sets. We complement this result by proving that for every $n \geq 18$ there exists a $2$-outerplanar $st$-digraph $G$ with $n$ vertices and a one-sided convex point set $S$ so that $G$ does not admit a 1-bend upward point-set embedding on $S$.

Figures (15)

• Figure 1: (a) Two edges that cross; (b) two edges that nest; (c) an example of a 2UBE; (d) an example of a 2UTBE; the bold edges have spine crossings, shown with small crosses; (e) removal of unnecessary sub-edges.
• Figure 2: Forbidden configurations.
• Figure 3: Impossible point sets.
• Figure 4: (a) A single-top 2UTBE $\gamma$; (b) the top-reduction $\gamma'_{top}$ of $\gamma$; (c) an enrichment of an UOSC point set $S$ (black squares) consistent with $\gamma$ with a good slope assignment. (d) A $1$-bend UPSE of $\gamma'_{top}$ on $S'$ computed as in \ref{['le:nice-assignment']}.
• Figure 5: (a) A $1$-bend UPSE of the graph of \ref{['fi:construction-a']} with one edge removed (the two end-vertices of the removed edge are highlighted); the top sub-edge of the removed edge covers only one vertex; (b) the addition of the missing edge (close-up).

Theorems & Definitions (22)

• lemma 1: $\star$
• proof
• lemma 2
• proof
• lemma 3: $\star$
• proof
• lemma 4: $\star$
• proof
• lemma 5: $\star$
• proof