Abstract
Graphs, the mathematical structures used to model pairwise relationships between objects, are pervasive across all scientific and technological fields. Graph drawing, the art of representing a graph in a visually intuitive way, stands out as one of the most captivating algorithmic challenges within graph theory. One of the most famous graph drawing algorithms is due to Walter Schnyder. At the heart of Schnyder’s algorithm lies a combinatorial construction called Schnyder wood, based on which the coordinates of the vertices are defined in a global manner.
The present work originates from an effort to generalize several previously known graph drawing algorithms with a similar flavor to Schnyder’s algorithm, which are due to He, Barri ́ere and Huemer, Bernardi and Fusy, and Fusy. We first extend the concept of Schnyder wood to a construction called “grand-Schnyder” structure and study its various combinatorial properties. We show that the grand-Schnyder framework unifies not only the classical Schnyder woods, but also several other constructions due to He, Bernardi and Fusy, and Felsner. We then present a graph drawing framework based on grand-Schnyder structures, which includes a straight-line drawing algorithm for plane maps with faces of degree 3 and 4 with no separating 3-cycles, and an orthogonal drawing algorithm for the dual of such plane maps. When specializing our algorithms to particular classes of plane maps we recover the aforementioned algorithms. We also present a barycentric graph drawing algorithm for 5-connected triangulations based on a further extension of grand-Schnyder structures.