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 $r\ge3$;
• 2) the degree of each internal vertex is $q$, where $q\ge3$, 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.