Consider a connected planar network with n vertices and m edges. Let f be the number of "faces" of the network, i.e., areas bounded by edges when the network is drawn in planar form. The "outside" of the network, the area extending to infinity on all sides, is also considered a face. The network can have multiedges and self-edges:
a) Write down the values of n, m, and f for a network with a single vertex and no edges.
b) How do n, m, and f change when we add a single vertex to the network along with a single edge attaching it to another vertex?
c) How do (n, m, and f change when we add a single edge between two extant vertices (or a self-edge attached to just one vertex), in such a way as to maintain the planarity of the network?
d) Hence by induction prove a general relation between n, m, and f for all connected planar networks.
e) Now suppose that our network is simple (i.e., it contains no multiedges or selfedges). Show that the mean degree c of such a network is strictly less than six.