Given two points on a Riemannian manifold, a geodesic connecting them is a critical point for the length functional (but not necessarily its minimum). Similarly, if we are given several points, one can define geodesic nets as critical points of the length functional on the space of graphs ``connecting" these points. This is equivalent to saying that all edges of the graph are geodesics, and at each new vertex of the graph the sum of all unit tangent vectors to the incident edges is equal to the zero vector.
While such geodesic nets are easy to define, not much is known about their classification, even on the Euclidean plane.
We discuss an easy to state question about geodesic nets, namely, if the total number of vertices on the net is bounded by the number of the points spanning the net. Here we are not allowing vertices of degree 2 as they do not change the geometry of the graph. We prove that a net spanned by three points on a non-positively curved plane has at most one extra vertex. This fact is new and has an unexpectedly difficult proof, even in the Euclidean case. We will also present work on examples of geodesics nets in the Euclidean plane spanned by 14 (or possibly just 7) points and having an arbitrary large number of vertices, suggesting that, in general, no such bound can exist.