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Remark on topological nature of upward planarity
Category Theory · Submitted 22 Oct 2020
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The main result is that two upward planar drawings of an acyclic directed graph are equivalent (connected by a deformation) if and only if they are connected by a planar isotopy which preserves the orientation and polarization of $G$.
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The notion of an upward plane graph in graph theory and that of a progressive plane graph (or plane string diagram) in category theory are essentially the same thing. In this paper, we combine the ideas in graph theory and category theory to explain why and in what sense upward planarity is a topological property. The main result is that two upward planar drawings of an acyclic directed graph are equivalent (connected by a deformation) if and only if they are connected by a planar isotopy which preserves the orientation and polarization of $G$. This result gives a positive answer to Selinger's conjectue, whose strategy is different from the solution recently given by Delpeuch and Vicary.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The notion of an upward plane graph in graph theory and that of a progressive plane graph (or plane string diagram) in category theory are essentially the same thing. | background 1.00 | 0.13 | 0.03 | asserted 1.00 |
| s1 | In this paper, we combine the ideas in graph theory and category theory to explain why and in what sense upward planarity is a topological property. | contribution 0.95 | 0.86 | 0.03 | asserted 0.92 |
| s2 | The main result is that two upward planar drawings of an acyclic directed graph are equivalent (connected by a deformation) if and only if they are connected by a planar isotopy which preserves the orientation and polarization of $G$. | result 1.00 | 0.91 | 0.98 | asserted 1.00 |
| s3 | This result gives a positive answer to Selinger's conjectue, whose strategy is different from the solution recently given by Delpeuch and Vicary. | interpretation 0.65 | 0.58 | 0.05 | asserted 1.00 |
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Original abstract
The notion of an upward plane graph in graph theory and that of a progressive plane graph (or plane string diagram) in category theory are essentially the same thing. In this paper, we combine the ideas in graph theory and category theory to explain why and in what sense upward planarity is a topological property. The main result is that two upward planar drawings of an acyclic directed graph are equivalent (connected by a deformation) if and only if they are connected by a planar isotopy which preserves the orientation and polarization of $G$. This result gives a positive answer to Selinger's conjectue, whose strategy is different from the solution recently given by Delpeuch and Vicary.
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This theorem also gives a positive answer to Selinger's conjectue [ref] with the strategy different from that of Delpeuch and Vicary [ref] .
Normalization for planar string diagrams and a quadratic equivalence algorithm · NPP-extension and main result · 0.86
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