Upward Pointset Embeddings of Planar st-Graphs

2024-08-30

International Symposium Graph Drawing and Network Visualization

AI Summary

This paper investigates the Upward Point-Set Embedding (UPSE) problem for planar st-graphs on a given point set: determining whether there exists a bijective mapping of vertices to prescribed points such that the straight-line drawing is both upward and plane. First, it establishes NP-completeness of UPSE recognition. Second, it presents a fixed-parameter algorithm parameterized by the st-cutwidth $k$, running in $O(n^{4k})$ time. Third, it devises an $O(n)$-delay algorithm to enumerate all UPSEs. Fourth, for st-cactus graphs (a generalization of st-cycles), it provides a necessary and sufficient condition for UPSE existence and an $O(n log n)$ recognition algorithm. Finally, it proves an equivalence between non-crossing monotone Hamiltonian cycles and UPSEs, enabling $O(n)$-delay enumeration thereof. The core contributions are: (i) settling the computational complexity of UPSE, (ii) delivering efficient exact algorithms—including FPT and output-sensitive enumeration—and (iii) uncovering structural properties that render UPSE tractable for restricted graph classes.

Technology Category

• Planning, Routing, and Scheduling: Planning under Uncertainty
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study upward pointset embeddings (UPSEs) of planar $st$-graphs. Let $G$ be a planar $st$-graph and let $S subset mathbb{R}^2$ be a pointset with $|S|= |V(G)|$. An UPSE of $G$ on $S$ is an upward planar straight-line drawing of $G$ that maps the vertices of $G$ to the points of $S$. We consider both the problem of testing the existence of an UPSE of $G$ on $S$ (UPSE Testing) and the problem of enumerating all UPSEs of $G$ on $S$. We prove that UPSE Testing is NP-complete even for $st$-graphs that consist of a set of directed $st$-paths sharing only $s$ and $t$. On the other hand, if $G$ is an $n$-vertex planar $st$-graph whose maximum $st$-cutset has size $k$, then UPSE Testing can be solved in $O(n^{4k})$ time with $O(n^{3k})$ space; also, all the UPSEs of $G$ on $S$ can be enumerated with $O(n)$ worst-case delay, using $O(k n^{4k} log n)$ space, after $O(k n^{4k} log n)$ set-up time. Moreover, for an $n$-vertex $st$-graph whose underlying graph is a cycle, we provide a necessary and sufficient condition for the existence of an UPSE on a given pointset, which can be tested in $O(n log n)$ time. Related to this result, we give an algorithm that, for a set $S$ of $n$ points, enumerates all the non-crossing monotone Hamiltonian cycles on $S$ with $O(n)$ worst-case delay, using $O(n^2)$ space, after $O(n^2)$ set-up time.

Problem

• Planar st-Graph
• Vertex Insertion
• Crossing-Free Drawing

Innovation

• UPSE Algorithm
• Complexity Optimization
• Ring Graph Processing