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.