Manipulating Weights to Improve Stress-Graph Drawings of 3-Connected Planar Graphs

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.