  Define a boundary point of a graph which is embedded in the Euclidean plane a
vertex which is incident to only one edge. In this paper we consider graphs
which are embedded in the Euclidean plane with a finite number of boundary
points. The simple geometric condition we impose on them is that the sum of
unit vectors of edges extending from each non-boundary vertex will be equal to
zero. We call such a graph a critical graph and ask to maximise the number of
vertices of critical graphs with a given size of boundary. The main results of
this paper give a sharp upper bound for the maximum number of vertices of
planar critical graphs, where the degree of each non-boundary vertex is 3 or 4.
