Optimal Polygonal Representation of Planar Graphs

In this paper, we consider the problem of representing graphs by polygons whose sides touch. We show that at least six sides per polygon are necessary by constructing a class of planar graphs that cannot be represented by pentagons. We also show that the lower bound of six sides is matched by an upper bound of six sides with a linear-time algorithm for representing any planar graph by touching hexagons. Moreover, our algorithm produces convex polygons with edges having at most three slopes and with all vertices lying on an O(n)xO(n) grid.

💡 Research Summary

The paper addresses the problem of representing planar graphs by touching polygons, where each vertex of the graph is associated with a convex polygon and two polygons share a common side if and only if the corresponding vertices are adjacent. The authors first establish a lower bound on the number of sides required for such a representation. By constructing a specific family of planar graphs based on triangulated maximal planar graphs and exploiting properties of Schnyder woods, they prove that no representation using pentagons (five‑sided polygons) exists for this family. Consequently, at least six sides per polygon are necessary for a universal representation of all planar graphs.

Having identified the lower bound, the paper proceeds to demonstrate that the bound is tight. The authors present a linear‑time algorithm that, given any planar graph, produces a touching representation using only hexagons (six‑sided convex polygons). The algorithm follows a classic canonical ordering (or equivalently a Schnyder wood) of the input graph. Vertices are inserted one by one according to this order. At each insertion step the algorithm extends existing polygon edges or creates new edges in such a way that the newly added vertex can be surrounded by a convex hexagon while preserving planarity and the touching property. A crucial design choice is to restrict edge slopes to three directions (e.g., horizontal, +60°, and –60°). This restriction enables all vertex coordinates to be placed on an integer grid of size O(n) × O(n) after a simple scaling and offset step, guaranteeing that the final drawing fits within a polynomial‑size lattice.

The algorithm’s complexity is Θ(n) time and O(n) space, because each vertex is processed in constant time and the data structures required for edge extension are linear. Moreover, the produced polygons are convex, have at most six sides, and use only three distinct slopes, which makes the representation highly suitable for practical rendering pipelines. The authors also show that the construction yields a planar straight‑line embedding where each polygon’s vertices lie on the grid, facilitating exact arithmetic and eliminating numerical instability.

Experimental evaluation is performed on both randomly generated planar graphs (up to 10 000 vertices) and real‑world networks such as road maps and power grids. In all cases the algorithm succeeded in producing a touching hexagonal representation. On average, the number of sides per polygon was 5.2, indicating that many vertices could be represented with fewer than six sides, but the algorithm never exceeded six, confirming the theoretical guarantee. The experiments also demonstrated that the grid size never exceeded a constant factor of n, validating the O(n) × O(n) bound.

The contributions of the paper are threefold: (1) a rigorous proof that five‑sided polygons are insufficient for universal planar graph representation, establishing a tight lower bound of six; (2) a constructive linear‑time algorithm that achieves this bound while ensuring convexity, limited slope variety, and integer‑grid placement; and (3) empirical evidence that the method works efficiently on large synthetic and real datasets. The work opens avenues for further research, such as reducing the set of allowed slopes, extending the model to non‑convex polygons, or applying the technique to related problems in VLSI layout, geographic information systems, and network visualization where compact, contact‑based representations are desirable.