Geodesic
A metric of constant curvature on polycycles
Matematičeskie zametki, Tome 78 (2005) no. 2, pp. 223-233

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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. 
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     author = {M. Deza and M. I. Shtogrin},
     title = {A metric of constant curvature on polycycles},
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     year = {2005},
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     url = {http://geodesic.mathdoc.fr/item/MZM_2005_78_2_a6/}
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M. Deza; M. I. Shtogrin. A metric of constant curvature on polycycles. Matematičeskie zametki, Tome 78 (2005) no. 2, pp. 223-233. http://geodesic.mathdoc.fr/item/MZM_2005_78_2_a6/