Drawing Reeb Graphs

TL;DR

The paper addresses the problem of visualizing Reeb graphs by minimizing edge crossings, proving NP-hardness for both straight-edge and curved-edge drawings. It introduces refined Reeb graphs that preserve crossing complexity and analyzes tractable acyclic cases (paths and caterpillars) as crossing-free instances, along with an optimal cycle-drawing algorithm that achieves at most $k-1$ crossings based on a top-down iteration measure. A key contribution is a reduction from Optimal Linear Arrangement using a triangular hexagonal grid gadget to establish NP-hardness, complemented by a detailed construction and correctness proof. The work lays a foundation for understanding Reeb graph visualization challenges and suggests directions for future research on more complex tree structures and higher-genus surfaces, with initial parameterized results by cycle count.

Abstract

Reeb graphs are simple topological descriptors with applications in many areas like topological data analysis and computational geometry. Despite their prevalence, visualization of Reeb graphs has received less attention. In this paper, we bridge an essential gap in the literature by exploring the complexity of drawing Reeb graphs. Specifically, we demonstrate that Reeb graph crossing number minimization is NP-hard, both for straight-lined and curved edges. On the other hand, we identify specific classes of Reeb graphs, namely paths and caterpillars, for which crossing-free drawings exist. We also give an optimal algorithm for drawing cycle-shaped Reeb graphs with the least number of crossings and provide initial observations on the complexities of drawing multi-cycle Reeb graphs. We hope that this work establishes the foundation for an understanding of the graph drawing challenges inherent in Reeb graph visualization and paves the way for future work in this area.

Figures (13)

• Figure 1: A two-manifold and its Reeb graph.
• Figure 2: Two level drawings of the same graph, one with $y$-monotone curves and one with straight line segments.
• Figure 4: A triangular hexagonal grid on the left and the same grid if we twist the structure vertically once over the second row.
• Figure 5: Graph on the left and its image based on an optimal order defined by a function $f: V \rightarrow \{1,2,...,|V|\}$ on the right.
• Figure 6: The corresponding Reeb graph construction of Fig. \ref{['fig:ex-graph-nphardproof']} in our proof. Black vertices are $V_1$ and red vertices are $V_2$. $E_1$ edges are shown in orange, $E_2$ edges are dotted, $E_3$ are in red and $E_4$ are the edges of the triangular hexagonal grids represented in black.

Theorems & Definitions (13)

• Definition 2: Refined Reeb Graph
• Lemma 3: Constructing a Refined Reeb Graph
• Lemma 4
• Lemma 5: Non-Stretchable Graph Drawing
• Theorem 6: RGCN is NP-Hard
• Lemma 7
• Lemma 8
• Definition 11: Top-Down Iteration Number
• Theorem 12: Cycle Drawing
• Lemma 13