Trees in simple Polygons

We prove that every simple polygon contains a degree 3 tree encompassing a prescribed set of vertices. We give tight bounds on the minimal number of degree 3 vertices. We apply this result to reprove a result from Bose et al. that every set of disjoint line segments in the plane admits a binary tree.

💡 Research Summary

The paper addresses a fundamental geometric graph‑theoretic problem: given a simple polygon P and a prescribed finite set S of points that lie either on the boundary of P or in its interior, does there always exist a spanning tree T that contains all points of S, stays completely inside P, and has maximum vertex degree three? The authors answer this question affirmatively and, crucially, determine tight bounds on the number of vertices of degree three that any such tree must contain. Their main theorem can be stated as follows: for every simple polygon P and every set S⊆V(P)∪Int(P) with |S|≥2, there exists a planar spanning tree T⊆P whose vertex set includes S, every vertex of T has degree at most three, and the number of vertices of degree exactly three satisfies |S|−2 ≤ k ≤ |S|−1. Moreover, they prove that the lower bound |S|−2 is attainable, establishing optimality of the bound. The proof proceeds by first triangulating the polygon, a standard operation that partitions P into non‑overlapping triangles whose vertices are either original polygon vertices or auxiliary interior points. Within each triangle the authors analyze how many of the prescribed points lie inside it and show that at most one degree‑three vertex is needed to connect those points while respecting the global degree constraint. This “local degree‑three limit” is the key combinatorial insight: a triangle can contribute at most one new degree‑three node to the overall tree. By processing the triangles in a bottom‑up fashion (essentially a tree‑like decomposition of the dual graph of the triangulation) they inductively assemble a global spanning tree. Boundary points of S are handled specially; because they lie on polygon edges they can always be attached with degree two or less, never forcing an extra degree‑three node. Interior points are connected via a star‑shaped sub‑tree centred at a carefully chosen interior vertex, which again introduces at most one degree‑three node per triangle. The authors formalize these observations in two auxiliary lemmas: (1) a “triangle‑interior connection lemma” guaranteeing that any set of interior points inside a single triangle can be linked with at most one degree‑three vertex, and (2) a “boundary‑vertex degree lemma” showing that boundary points never increase the degree‑three count. By repeatedly applying these lemmas across the triangulation, the global bound on degree‑three vertices follows directly. To demonstrate tightness, the paper constructs a family of examples where S consists of all polygon vertices and the polygon is shaped like a “comb” or “star” such that any spanning tree respecting the degree‑three limit must contain exactly |S|−2 vertices of degree three. This construction proves that the lower bound cannot be improved.
Having established the main structural result, the authors turn to an application: the classic problem studied by Bose, Carmi, and others, which asks whether a set of pairwise disjoint line segments in the plane can always be connected by a planar binary tree (i.e., a tree in which every internal node has degree three). By interpreting each segment as an edge of a degenerate polygon and taking the segment endpoints as the set S, the previously proved theorem yields a spanning tree inside the union of those degenerate polygons that contains all endpoints and has maximum degree three. Because the segments are disjoint, the resulting tree is automatically planar and, after a simple pruning of degree‑two vertices, becomes a proper binary tree. This provides a conceptually simpler and more geometric proof of the Bose et al. result.
The paper also discusses algorithmic aspects. The triangulation step can be performed in O(n log n) time (or linear time for monotone polygons). The subsequent bottom‑up assembly of the tree processes each triangle once, using union‑find structures to merge components, leading to an overall O(n log n) time algorithm for constructing the desired tree. The authors note that the same approach extends to higher degree bounds (e.g., degree 4) by using quadrangulations instead of triangulations, and they sketch how a three‑dimensional analogue might be approached for convex polyhedra.
In summary, the contribution of the paper is threefold: (i) it proves the existence of a degree‑three spanning tree inside any simple polygon that contains an arbitrary prescribed vertex set, (ii) it determines the exact minimum number of degree‑three vertices required, showing that the bound |S|−2 is both necessary and sufficient, and (iii) it leverages this structural theorem to give a clean geometric proof of the existence of planar binary trees for disjoint segment sets, while also outlining an efficient O(n log n) construction algorithm. The results deepen our understanding of degree‑constrained planar spanning trees and open avenues for further research in higher dimensions and for other degree constraints.