[Paper Review] How Much Work Does It Take To Straighten a Plane Graph Out?

Mihyun Kang, Mathias Schacht+1 more|ArXiv.org|Jul 23, 2007

Computational Geometry and Mesh Generation10 references3 citations

TL;DR

This paper proves that transforming a plane graph into a straight-line drawing requires moving nearly all vertices in the worst case, establishing that the minimum number of vertices that must be shifted, denoted s(n), satisfies s(n) = n(1 - o(1)). The result is derived by constructing a 3-connected planar graph with vertices arranged in nested, non-overlapping triangulated clusters on a circle, showing that only O(n) vertices can remain fixed in any straight-line embedding due to geometric constraints on disjoint triangular regions.

ABSTRACT

We prove that if one wants to make a plane graph drawing straight-line then in the worst case one has to move almost all vertices.

Motivation & Objective

• To determine the minimum number of vertices that must be moved to convert any plane graph into a straight-line drawing.
• To establish a lower bound on the number of vertices that must be shifted in the worst-case scenario for planar graphs.
• To analyze the structural constraints that limit the number of fixed vertices in straight-line embeddings of planar graphs.
• To construct explicit families of 3-connected planar graphs where almost all vertices must be relocated to achieve a crossing-free straight-line drawing.
• To connect the problem to the Planarity Game, where optimal play requires shifting nearly all vertices in the worst case.

Proposed method

• Construct a graph G with n = k(s + k) vertices divided into s + k disjoint sets V_i, each of size k.
• Embed G such that each V_i induces a maximal planar (triangulated) subgraph, and the subgraphs are placed in mutually disjoint outer faces.
• Ensure global 3-connectivity by adding inter-cluster edges between consecutive V_i and a final edge closing the cycle.
• Define a 'bad' circular drawing where all V_i lie on a circle at vertices of a regular k-gon, creating edge crossings.
• Analyze any straight-line crossing-free drawing G' of G by associating each V'_i with a triangular face T_i in the convex hull of G'[V'_i].
• Use geometric disjointness of T_i (except possibly one) and the absence of alternating 4-tuples (xyxy) in circular label sequences to bound the number of fixed vertices.

Experimental results

Research questions

• RQ1 What is the worst-case number of vertices that must be moved to convert a plane graph into a straight-line drawing?
• RQ2 Can the number of fixed vertices in such a transformation be bounded below by a sublinear function of n?
• RQ3 How does the geometric arrangement of vertex clusters affect the maximum number of vertices that can remain fixed?
• RQ4 To what extent do 3-connectivity and planarity constraints limit the number of fixed vertices in straight-line embeddings?
• RQ5 How does this problem relate to the Planarity Game, and what is the minimal number of moves required in the worst-case game instance?

Key findings

• The paper establishes that s(n) = n(1 - o(1)), meaning that in the worst case, almost all vertices must be moved to achieve a straight-line drawing.
• An explicit lower bound is derived: s(n) ≥ n - 2√n - 1, showing that at least this many vertices must be shifted.
• The construction ensures that at most k - 1 of the vertex sets V_i can have two or more vertices fixed in position, due to pairwise disjoint triangular regions in the straight-line embedding.
• The bound is tight for graphs with vertices in weakly convex position, as the result holds when vertices are placed at any prescribed set of n points in such position.
• The analysis shows that only O(k) = O(√n) vertices can be fixed in the worst case, confirming that sublinear fixed sets are unavoidable.
• The result improves upon prior bounds for specific graph classes like fans and star forests, where fix(G) ≤ 3√n - 3, and extends to general 3-connected planar graphs.