We study vertex-labeled graphs that can be embedded on a given point set such that every edge is a polyline with $k$ bends per edge, where $k\in \mathbb{N}$. It is shown that on every $n$-element point set in the plane, at most $\exp(O(n\log(2+k)))$ labeled graphs can be embedded using polyline edges with $k$ bends per edge, and this bound is the best possible. This is the first exponential upper bound for the number of labeled plane graphs where the edges are polylines of constant complexity. Standard tools developed for the enumeration of straight-line graphs, such as triangulations and crossing numbers, do not seem applicable in this scenario. Furthermore, the exponential upper bound does not carry over to other popular relaxations of straight-line edges; for example, the number of labeled planar graphs that admit an embedding with $x$-monotone edges on $n$ points is superexponential.

It is shown that on every n-element point set in the plane, at most exp(O(kn)) labeled planar graphs can be embedded using polyline edges with k bends per edge. This is the first exponential upper bound for the number of labeled plane graphs where the edges are polylines of constant size. Several standard tools developed for the enumeration of straight-line graphs, such as triangulations and crossing numbers, are unavailable in this scenario. Furthermore, the exponential upper bound does not carry over to other popular relaxations of straight-line edges: for example, the number of plane graphs realizable with x-monotone edges on n points is already super-exponential.

We show that it is an NP-hard problem to decide for a given set P of n points in the Euclidean plane and a given parameter k∈R, whether P admits a spanning tree of maximum vertex degree four whose sum of edge lengths does not exceed k.