## 14.2 Kernel Elementary Polycycles

Call *kernel Ker*(*P*) of an(*R*, *q*)_{simp}-polycycle *P* the cell complex formed by its vertices, edge, and faces that do not contain a boundary vertex. Call an (*R*, *q*) polycycle *kernel-elementary* if it is an *r*-gon or if it has a nonempty 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).

**Theorem 14.2**

*(i) If an* (*R*, *q*)_{simp}-polycycle is kernel-elementary, then it is elementary.

*(ii) If* (*R*, *q*) *is elliptic, then any elementary* (*R*, *q*)_{simp}-polycycle is also kernelelementary.

*Proof*. (i) Take a kernel-elementary (*R*, *q*)-polycycle *P*; one can assume it to be different from an *r*-gon. Let *P*_{1},..., *P*_{m} be the elementary components of this polycycle. The connectedness condition on the kernel gives that all *P*_{i} but one are *r*-gons with *r* ∈ *R*. But removing the components *P*_{i} that are *r*-gons does not change the kernel; thus, *m* =1and *P* is elementary.

(ii) Consider any two vertices of an *r*-gon of an elliptic (*R*, *q*)-polycycle that belongs to the kernel of this polycycle. The shortest edge path between these vertices lies inside the union of two stars of *r*-gons with the centers at these two vertices; this result can easily be verified in each particular case for any elliptic parameters (*R*, *q*) = ({2, 3, 4, 5},3), ({2, 3},4), and ({2, 3},5). Hence, any *r*-gon of an elliptic (*R*, *q*)-polycycle is incident with only one simply connected component of its kernel. All *r*-gons that are incident with ...