Untangling polygons and graphs

Untangling is a process in which some vertices of a planar graph are moved to obtain a straight-line plane drawing. The aim is to move as few vertices as possible. We present an algorithm that untangles the cycle graph C_n while keeping at least \Omega(n^{2/3}) vertices fixed. For any graph G, we also present an upper bound on the number of fixed vertices in the worst case. The bound is a function of the number of vertices, maximum degree and diameter of G. One of its consequences is the upper bound O((n log n)^{2/3}) for all 3-vertex-connected planar graphs.

💡 Research Summary

The paper studies the “untangling” problem for planar graphs, which asks how many vertices can remain at their original positions when the graph is redrawn with straight‑line edges so that no two edges cross. Formally, for a mapping δ: V(G)→ℝ² the quantity fix(G, δ) is the maximum number of vertices that can stay fixed in any plane straight‑line drawing that respects δ; fix(G) is the minimum of fix(G, δ) over all possible δ.

Previous work gave a lower bound Ω(√n) and an upper bound O((n log n)^{2/3}) for the cycle graph Cₙ, and only a very weak Ω(n^{1/4}) lower bound for general planar graphs. The authors close the gap for cycles and provide a unified upper bound for all planar graphs that depends on the number of vertices n, the maximum degree Δ, and the diameter diam.

Cycle graph algorithm.
The authors fix a horizontal direction so that no two vertices share a horizontal or vertical line. They take the highest m≈n−O(n^{2/3}) vertices and partition them into l layers, each containing s² vertices where s=2l and l≈(m/16)^{1/3}. In each layer they apply the Erdős‑Szekeres lemma to extract a monotone subsequence of exactly s vertices (indices either strictly increasing or decreasing). These s vertices become the “fixed” vertices of that layer; the remaining vertices of the layer and all vertices within graph‑distance ≤2(l−i−1) of any selected vertex are declared “free”. By counting carefully they show that the total number of fixed vertices is ls=2l²≥(2/5)n^{2/3}−O(n^{1/3}), i.e. fix(Cₙ)=Ω(n^{2/3}). The construction then places the free vertices on a staircase of horizontal and vertical segments that connect the fixed vertices without creating crossings. This algorithm is deterministic and works for any initial placement of the cycle’s vertices.

General planar‑graph upper bound.
The second part builds on a probabilistic crossing‑number bound due to Pach and Tardos. For a given set of t vertices that we would like to keep fixed, the authors consider two edge sets: a red cycle formed by the convex hull of the fixed vertices, and a black Hamiltonian cycle obtained by a random permutation of the same vertices (the “DFS‑cycle” of a spanning tree). Lemma 2 gives an upper bound on the probability that the combined graph has at most K crossings, where K depends on t, n, and a parameter D derived from K.

Applying this to a spanning tree T of the planar graph G, they define

 t = ⌈300 √n log n (√Δ + min{6 n/(log²n), √diam})⌉

and set K accordingly. By choosing D≈t log n, the probability that any t‑tuple of vertices can be kept fixed while preserving planarity becomes less than 1. Hence for every drawing δ there exists a set of at most t vertices that must move, which yields

 fix(G) ≤ 300 √n log n (√Δ + min{6 n/(log²n), √diam}).

Special cases.

• For any 3‑vertex‑connected planar graph, Δ and diam are constants, so the bound simplifies to O((n log n)^{2/3}).
• If both Δ and diam are O(log n), the bound becomes O(√n (log n)^{3/2}), which is close to the best known lower bound Ω(√n).
• If a graph satisfies fix(G) ≥ ε n for a constant ε, the bound forces the existence of a vertex of degree at least c n^{ε²}/log²n, showing that a high fraction of fixed vertices can only occur in graphs with very high degree vertices.

Proof techniques.
The cycle algorithm relies on geometric ordering and the Erdős‑Szekeres monotone subsequence theorem to guarantee a large monotone block in each layer. The general bound uses a combination of depth‑first‑search (DFS) cycles, convex‑hull red cycles, and a probabilistic crossing‑number analysis. The authors also show that any planar graph has a spanning tree of diameter at most twice the graph’s diameter (via BFS), which allows the use of the tree‑based analysis for arbitrary planar graphs.

Conclusion.
The paper makes two major contributions: (1) a deterministic algorithm that fixes Ω(n^{2/3}) vertices of a cycle, dramatically narrowing the previous Ω(√n)–O((n log n)^{2/3}) gap; (2) a unified upper bound on fix(G) for all planar graphs expressed in terms of n, Δ, and diam, from which several corollaries follow, including the O((n log n)^{2/3}) bound for 3‑vertex‑connected planar graphs. These results advance the theoretical understanding of untangling and open avenues for tighter bounds, extensions to non‑planar settings, and practical algorithms for graph drawing applications.