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A recent data visualization literacy study shows that most people cannot read networks that use hierarchical cluster representations such as “supernoding” and “edge bundling.” Other studies that compare standard node-link representations with map-like visualizations show that map-like visualizations are superior in terms of task performance, memorization and engagement. With this in mind, we propose the Zoomable Multi-Level Tree (ZMLT) algorithm for maplike visualization of large graphs that is representative, real, persistent, overlapfree labeled, planar, and compact. These six desirable properties are formalized with the following guarantees: (1) The abstract and embedded trees represent the underlying graph appropriately at different level of details (in terms of the structure of the graph as well as the embedding thereof); (2) At every level of detail we show real vertices and real paths from the underlying graph; (3) If any node or edge appears in a given level, then they also appear in all deeper levels; (4) All nodes at the current level and higher levels are labeled and there are no label overlaps; (5) There are no crossings on any level; (6) The drawing area is proportional to the total area of the labels. This algorithm is implemented and we have a functional prototype for the interactive interface in a web browser.
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This content will become publicly available on November 14, 2025
Grand-Schnyder Woods
We define a far-reaching generalization of Schnyder woods which encompasses many classical combinatorial structures on planar graphs. Schnyder woods are defined for planar triangulations as certain triples of spanning trees covering the triangulation and crossing each other in an orderly fashion. They are of theo- retical and practical importance, as they are central to the proof that the order dimension of any planar graph is at most 3, and they are also underlying an elegant drawing algorithm. In this article we extend the concept of Schnyder wood well beyond its original setting: for any integer d ≥ 3 we define a “grand-Schnyder” structure for (embedded) planar graphs which have faces of degree at most d and non-facial cycles of length at least d. We prove the existence of grand-Schnyder structures, provide a linear construction algorithm, describe 4 different incarnations (in terms of tuples of trees, corner labelings, weighted orientations, and marked orientations), and define a lattice for the set of grand Schnyder structures of a given planar graph. We show that the grand-Schnyder framework unifies and extends several clas- sical constructions: Schnyder woods and Schnyder decompositions, regular edge-labelings (a.k.a. transversal structures), and Felsner woods.
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- Award ID(s):
- 2154242
- PAR ID:
- 10588078
- Publisher / Repository:
- Springer
- Date Published:
- Journal Name:
- Annals of Combinatorics
- ISSN:
- 0218-0006
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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