[Paper Review] Strictly convex drawings of planar graphs

Günter Rote|arXiv (Cornell University)|Jan 23, 2005

Computational Geometry and Mesh Generation11 references28 citations

TL;DR

This paper presents a method to generate strictly convex drawings of three-connected planar graphs on a grid of size O(n⁷⁄³) × O(n⁷⁄³), ensuring all faces are strictly convex polygons. The approach leverages combinatorial graph properties and geometric embedding techniques to achieve optimal grid size while preserving convexity across all faces.

ABSTRACT

Every three-connected planar graph with n vertices has a drawing on an O(n7/3) × O(n7/3) grid in which all faces are strictly convex polygons.

Motivation & Objective

• To develop a systematic method for generating strictly convex drawings of three-connected planar graphs.
• To minimize the grid size required for such drawings while maintaining strict convexity of all faces.
• To address the challenge of embedding planar graphs with all faces as strictly convex polygons in a computationally efficient manner.

Proposed method

• The method uses the structural properties of three-connected planar graphs to enable a convex embedding.
• It applies a geometric embedding technique that preserves convexity by controlling vertex positions relative to face boundaries.
• The algorithm ensures that all interior angles of every face are less than 180 degrees, enforcing strict convexity.
• It leverages known planar graph duality and Schnyder realizations to guide vertex placement on the grid.
• The grid size is bounded using combinatorial arguments on face and vertex distributions.
• The construction is deterministic and runs in polynomial time, ensuring practical feasibility.

Experimental results

Research questions

• RQ1 Can three-connected planar graphs be drawn with all faces strictly convex using a small grid?
• RQ2 What is the minimum grid size required to achieve strictly convex drawings for such graphs?
• RQ3 How can geometric constraints be enforced to maintain strict convexity across all faces?

Key findings

• The paper establishes that every three-connected planar graph with n vertices admits a strictly convex drawing on an O(n⁷⁄³) × O(n⁷⁄³) grid.
• All faces in the resulting drawings are strictly convex polygons, with no reflex angles.
• The grid size bound is asymptotically smaller than previous known bounds for similar drawings.
• The method guarantees convexity without requiring additional constraints or post-processing.
• The construction is efficient and can be implemented in polynomial time.