Complexity of Planar Embeddability of Trees inside Simple Polygons

Geometric embedding of graphs in a point set in the plane is a well known problem. In this paper, the complexity of a variant of this problem, where the point set is bounded by a simple polygon, is considered. Given a point set in the plane bounded by a simple polygon and a free tree, we show that deciding whether there is a planar straight-line embedding of the tree on the point set inside the simple polygon is NP-complete. This implies that the straight-line constrained point-set embedding of trees is also NP-complete, which was posed as an open problem in [8].

💡 Research Summary

The paper investigates a geometric graph‑embedding problem that lies at the intersection of computational geometry and graph theory. The classic point‑set embedding problem asks whether a given graph can be drawn with straight‑line edges on a prescribed set of points in the plane, typically assuming that the points are unrestricted. For trees, polynomial‑time algorithms are known in this unrestricted setting. However, many practical scenarios impose a spatial boundary: the points must lie inside a simple polygon that represents a physical workspace, a building floorplan, or any other confined region. The authors formalize this constrained version as follows: given a simple polygon P, a set S of points lying inside P, and a free tree T with |T| = |S|, decide whether there exists a bijection between the vertices of T and the points of S such that every edge of T is drawn as a straight‑line segment connecting its two assigned points, the segment stays completely inside P, and no two segments intersect.

The main contribution is a proof that this decision problem is NP‑complete. Membership in NP is straightforward: a candidate mapping can be verified in polynomial time by checking that each edge lies inside P and that the set of edges is pairwise non‑crossing. The hardness proof proceeds by a polynomial‑time reduction from PLANAR 3‑SAT, a canonical NP‑complete problem. The reduction constructs three types of gadgets inside a carefully designed polygon:

1. Variable gadgets – small sub‑polygons each containing a set of points that admit exactly two feasible embeddings of a tiny subtree, corresponding to the logical values true and false. The geometry forces the subtree to choose one of the two configurations and prevents any other placement.

2. Clause gadgets – structures that connect to the three variable gadgets appearing in a clause. The clause gadget contains a subtree that can be embedded without crossing only if at least one of the incident variable gadgets is in the “true” configuration. This mirrors the satisfaction condition of a clause.

3. Global spanning tree – a backbone that links all variable and clause gadgets into a single tree. Its edges are drawn so that they must follow prescribed corridors inside the polygon, ensuring that the local choices made in the gadgets are globally consistent.

Two key lemmas are proved: (i) each variable gadget enforces a binary choice, and (ii) each clause gadget is realizable if and only if the associated Boolean clause is satisfied. By assembling the gadgets according to the planar embedding of the 3‑SAT formula, the authors obtain a polygon P, a point set S, and a tree T such that the tree can be embedded inside P on S without crossings exactly when the original formula is satisfiable. The construction is linear in the size of the formula, guaranteeing a polynomial‑time reduction.

Consequently, the constrained point‑set embedding problem for trees is NP‑complete. This resolves an open question posed in earlier work (reference