Some Euler-type formulas for planar graphs

where V is the number of vertices, E is the number of edges, and F is the number of faces in the given graph, including the exterior face. This formula corresponds to the special case g = 0 (simple connectedness) of the more general Poincaré formula for genus g surfaces, in which χ ≡ χ(g) = 2− 2g. In this note we derive several analytical relations, similar to Euler’s formula, which hold for some classes of planar graphs that we introduce below. A topological graph is a graph drawn in the plane such that its vertices are represented by points and its edges are represented by arcs connecting the corresponding points such that no two arcs intersect except at a common endpoint. The classes of graphs considered here are defined as follows.