Kernel-elementary polycycle

Call an (r, g)-polycycle kernel-elementary if it is an r-gon or if it has a non-empty connected kernel, such that the deletion of any face from the kernel will diminish it (i.e. any face of the polycycle is incident to its kernel). 

Proof, (i) Take a kernel-elementary (r, )-polycycle P we can assume it to be different from an r-gon. Let Pi,..., Pr be the elementary components of this polycycle. The connectedness condition on the kernel gives that all P but one are r-gons. But removing the P,- that are r-gons does not change the kernel so, r = 1 and P is elementary. 

In [DeSt02b] the notion of kernel-elementary was called elementary. See below for an example of a (6, 3)-polycycle, which is elementary but not kernel-elementary (since its kernel is not connected) ... 

The decomposition Theorem 7.1.1 (of (r, elementary polycycles) is the main reason why we prefer the property to be elementary to kernel-elementary. Another reason is that if an (r, g)g(m-polycycle is elementary, then its universal cover is also elementary. However, the notion of kernel elementariness will be useful in the classification of infinite elementary ( 3,4, 5, 3)- and ( 2, 3, 5)-polycycles. 

We can now classify all kernel-elementary elliptic (r, q)gen-polycycles obtained in [DDS05b], Such a polycycle is either elementary or it is obtained by self-gluing of an elementary (r, <7>-polycycle. Hence, the list is as follows ... 

All kernel-elementary (5,3)gen-polycycles, obtained by self-gluing of elementary (5,3)-polycycles, are E 0... 

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