A census of graph-drawing algorithms based on generalized transversal structures

TL;DR

The paper introduces two linear-time graph-drawing algorithms based on a unified grand-Schnyder framework (4-GS woods) that generalizes transversal structures and separating decompositions. It presents a straight-line drawing algorithm for 3,4-angulations and a dual orthogonal drawing algorithm, each with face-counting and increasing-function variants, and provides rigorous planarity proofs, grid-size analyses, and optimizations. By showing how classical algorithms (Fu07b, He93, Barrière-Huemer, Bernardi–Fusy) arise as special cases, the work unifies a broad family of plane-graph drawings under a single combinatorial structure, and also discusses bend-minimization via Tamassia flows. The methods yield linear-time construction, offer compact grid bounds, and enable extensions to graphs with degree-2 vertices, providing both theoretical insight and practical drawing tools. Overall, the paper delivers a cohesive, scalable framework that connects and extends key graph-drawing algorithms through 4-GS theory, with implications for efficient and compact planar representations.

Abstract

We present two graph drawing algorithms based on the recently defined "grand-Schnyder woods", which are a far-reaching generalization of the classical Schnyder woods. The first is a straight-line drawing algorithm for plane graphs with faces of degree 3 and 4 with no separating 3-cycle, while the second is a rectangular drawing algorithm for the dual of such plane graphs. In our algorithms, the coordinates of the vertices are defined in a global manner, based on the underlying grand-Schnyder woods. The grand-Schnyder woods and drawings are computed in linear time. When specializing our algorithms to special classes of plane graphs, we recover the following known algorithms: (1) He's algorithm for rectangular drawing of 3-valent plane graphs, based on transversal structures, (2) Fusy's algorithm for the straight-line drawing of triangulations of the square, based on transversal structures, (3) Bernardi and Fusy's algorithm for the orthogonal drawing of 4-valent plane graphs, based on 2-orientations, (4) Barriere and Huemer's algorithm for the straight-line drawing of quadrangulations, based on separating decompositions. Our contributions therefore provide a unifying perspective on a large family of graph drawing algorithms that were originally defined on different classes of plane graphs and were based on seemingly different combinatorial structures.

Figures (44)

• Figure 1: Straight-line drawing of a plane graph with faces of degree 3 and 4 (left), and rectangular drawing of its dual (right).
• Figure 2: Left: The local conditions (at inner vertices and at outer vertices) for transversal structures. Right: Example of a transversal structure on a triangulation of the square.
• Figure 3: Left: The local conditions, at inner vertices and at outer vertices, for separating decompositions. Right: Example of a separating decomposition on a simple quadrangulation.
• Figure 4: The straight-line drawing obtained using the transversal structure shown in Figure \ref{['fig:transversal']}. The blue and red separating paths are shown for the circled vertex $v$. For this vertex, the blue path $P_b(v)$ has 4 faces on its right in the blue map, and the red path has 5 faces on its left in the red map. Accordingly, $v$ has coordinates $(5,4)$ in the drawing. Placing similarly all the inner vertices yields the planar straight-line drawing shown on the right. Since the red and blue maps have 10 and 9 inner faces respectively, the grid has size $10\times 9$.
• Figure 5: The straight-line drawing obtained using the separating decomposition shown in Figure \ref{['fig:separating_decomp']}. The even (resp. odd) separating path of the circled vertex is shown, with 5 faces on its left (resp. $7$ faces on its right). Accordingly the circled vertex has coordinates $(5,7)$ in the drawing. Placing similarly all the inner vertices yields the planar straight-line drawing shown on the right. Since there are 11 inner faces, the grid has size $11\times 11$.

Theorems & Definitions (60)

• Definition 2.1
• Remark 2.2
• Remark 2.3
• Lemma 2.4
• proof
• Definition 2.5
• Theorem 3.1
• Remark 3.2
• Theorem 3.3
• Example 3.4