A metric of constant curvature on polycycles

We prove the following main theorem of the theory of (r, q)-polycycles. Suppose a nonseparable plane graph satisfies the following two conditions: (1) each internal face is an r-gon, where r >= 3 (2) the degree of each internal vertex is q, where q >= 3, and the degree of each boundary vertex is at most q and at least 2. Then it also possesses the following third property: (3) the vertices, the edges, and the internal faces form a cell complex. Simple examples show that conditions (1) and (2) are independent even provided condition (3) is satisfied. These are the defining conditions for an (r, q)-polycycle.