Totally elementary polycycle

An (R, 3)-polycycle is called totally elementary if it is elementary and if, after removing any face adjacent to a hole, we obtain a non-elementary (R, 3)-polycycle. So, an elementary (R, 3)-polycycle is totally elementary if and only if it is not the result of an extension of some elementary (R, 3)-polycycle. See below for an illustration of this notion ... 

All totally elementary ( 3,4,5, 3)-polycycles are enumerated in Lemma 7.4.3. We will now classify all finite elementary ( 3,4,5, 3)-polycycles with one hole. Such a polycycle with N interior faces is either totally elementary, or it is obtained from another such polycycle with N — 1 interior faces by addition of a face. 

There is no elementary ( 3,4,5), 3)-polycycles with 2 interior faces so, all elementary ( 3,4,5), 3)-polycycles with 3 interior faces are totally elementary and, by Lemma 7.4.3, we know them. Also, by Lemma 7.4.3, there are no finite totally elementary ( 3,4,5, 3)-polycycles with more than 3 interior faces. We iterate the following procedure starting at N = 3 ... 

Proof We need to prove only that there are no more than these n + 1 fillings. First, since all runs of two are 22, the only possible elementary (5,3)-polycycles are E, Ci, or C3. Secondly, since the (5,3)-boundary has exactly 4 runs of 2, the total number of (5, 3)-polycycles Ei and C3 is 2. 

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