Abstract
We study methods to manipulate weights in stress-graph embeddings to improve convex straight-line planar drawings of 3-connected planar graphs. Stress-graph embeddings are weighted versions of Tutte embeddings, where solving a linear system places vertices at a minimum-energy configuration for a system of springs. A major drawback of the unweighted Tutte embedding is that it often results in drawings with exponential area. We present a number of approaches for choosing better weights. One approach constructs weights (in linear time) that uniformly spread all vertices in a chosen direction, such as parallel to the x- or y-axis. A second approach morphs x- and y-spread drawings to produce a more aesthetically pleasing and uncluttered drawing. We further explore a “kaleidoscope” paradigm for this xy-morph approach, where we rotate the coordinate axes so as to find the best spreads and morphs. A third approach chooses the weight of each edge according to its depth in a spanning tree rooted at the outer vertices, such as a Schnyder wood or BFS tree, in order to pull vertices closer to the boundary.
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Notes
- 1.
Proofs of Fáry’s Theorem, that any simple, planar graph can be embedded in the plane without crossings so each edge is drawn as a straight line segment, came earlier [7, 15, 17], but these proofs do not give specific coordinates for the vertices; hence, it is not clear they can be called “graph drawing algorithms.”.
- 2.
Tutte’s approach can be viewed as being for the case when \(w_{u,v}=1\) for each edge.
- 3.
However, their proof is only valid for polyhedra that have a triangle face.
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This research was supported in part by NSF grant CCF-2212129.
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Chiu, A., Eppstein, D., Goodrich, M.T. (2023). Manipulating Weights to Improve Stress-Graph Drawings of 3-Connected Planar Graphs. In: Bekos, M.A., Chimani, M. (eds) Graph Drawing and Network Visualization. GD 2023. Lecture Notes in Computer Science, vol 14466. Springer, Cham. https://doi.org/10.1007/978-3-031-49275-4_10
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