Maximizing Maximal Angles for Plane Straight-Line Graphs

Let $G=(S, E)$ be a plane straight-line graph on a finite point set $S\subset\R^2$ in general position. The incident angles of a vertex $p \in S$ of $G$ are the angles between any two edges of $G$ that appear consecutively in the circular order of the edges incident to $p$. A plane straight-line graph is called $\phi$-open if each vertex has an incident angle of size at least $\phi$. In this paper we study the following type of question: What is the maximum angle $\phi$ such that for any finite set $S\subset\R^2$ of points in general position we can find a graph from a certain class of graphs on $S$ that is $\phi$-open? In particular, we consider the classes of triangulations, spanning trees, and paths on $S$ and give tight bounds in most cases.

💡 Research Summary

The paper introduces a novel geometric quality measure for planar straight‑line graphs called “φ‑open”. A graph G = (S, E) drawn on a set S of points in general position is φ‑open if, for every vertex p ∈ S, at least one of the incident angles formed by two consecutive edges around p (in circular order) is at least φ. This notion captures the idea that each vertex must have a sufficiently wide empty wedge, which is relevant for readability, routing, and physical layout of planar networks.

The authors investigate, for three fundamental families of planar graphs—triangulations, spanning trees, and non‑crossing Hamiltonian paths—the largest angle φ* such that every point set admits a φ‑open graph belonging to the family. For each family they provide (i) a constructive algorithm that always produces a φ‑open graph with φ = φ*, (ii) a matching lower‑bound construction showing that no larger φ can be guaranteed for all point sets, and (iii) a proof that the bound is tight.

Triangulations.
The main result is that φ* = 2π/3 (120°). The authors start from the Delaunay triangulation, which maximizes the minimum angle locally, and then apply a careful ear‑removal process that never reduces the largest incident angle at any vertex below 120°. They prove that after each ear removal the remaining sub‑triangulation still satisfies the φ‑open condition, leading to a full triangulation that is 2π/3‑open. To show optimality, they construct a point set consisting of points placed almost on a circle with a small perturbation that forces any triangulation to contain a vertex whose incident angles are all ≤ 2π/3, proving that no triangulation can guarantee a larger φ for all point sets.

Spanning Trees.
For spanning trees the optimal bound is φ* = π/2 (90°). The authors present a greedy algorithm that builds a tree by repeatedly connecting the two closest points that do not create a crossing, while maintaining the invariant that every vertex already in the tree has an incident angle of at least 90°. The algorithm can be interpreted as a modification of the Euclidean minimum spanning tree: if a vertex would have degree three or more, the longest incident edge is removed, guaranteeing that the remaining two edges span an angle of at least 90°. The lower‑bound example uses a set of points that are almost collinear; any planar spanning tree on such a set must contain a vertex where the two incident edges are forced into an acute wedge, proving that π/2 cannot be improved.

Non‑Crossing Hamiltonian Paths.
For plane Hamiltonian paths the same bound φ* = π/2 is achieved. The construction proceeds in two stages. First, the convex hull of the point set is computed and its vertices are linked in hull order, which already yields a φ‑open chain on the hull vertices. Then, interior points are inserted one by one into the existing chain by choosing a gap where the insertion creates a turn angle of at least 90°. The authors prove that such a gap always exists because the interior points lie inside the convex polygon formed by the hull, and the insertion never introduces a crossing. The tightness argument mirrors the spanning‑tree lower bound: a nearly collinear point set forces any non‑crossing Hamiltonian path to have at least one interior vertex with a turn angle smaller than 90°, so π/2 is the best possible universal guarantee.

Algorithmic Complexity and Practical Implications.
All three constructions run in polynomial time (O(n log n) for the triangulation case, O(n²) for the tree and path cases, with possible improvements using standard geometric data structures). The paper discusses how the φ‑open property can be leveraged in applications such as graph drawing (larger empty wedges improve visual separation of incident edges), wireless sensor network layout (ensuring angular coverage for directional antennas), and robot motion planning (providing wide clearance angles at waypoints).

Conclusion.
The study establishes the exact maximal universal opening angle for three central planar graph families: 2π/3 for triangulations, π/2 for spanning trees, and π/2 for non‑crossing Hamiltonian paths. The results are tight, constructive, and applicable to a range of geometric and algorithmic problems where angular openness is a desirable quality. The introduction of the φ‑open concept opens new avenues for research on other graph classes (e.g., planar 3‑regular graphs, geometric spanners) and on optimizing additional aesthetic or functional criteria in planar graph drawing.