## 14.3 Classification of Elementary ({2, 3, 4, 5}, 3)-Polycycles

For 2 ≤ *m* ≤ ∞, denote by *Barrel*_{m} the ({5}, 3)-polycycle with two *m*-gonal holes separated by two *m*-rings of 5-gonal proper faces. This polycycle is nonextensible if and only if *m* ≥ 6. Its symmetry group, *D*_{md}, coincideswith the symmetry group of the underlying 3-valent plane graph if and only if *m* ≠ 5. See below for pictures of *Barrel*_{m} for *m* = 2, 3, 4, 5, and ∞:

**Theorem 14.3** *The list of elementary* ({2, 3, 4, 5},3)*-polycycles consists of*:

*(i) 204 **sporadic* ({2, 3, 4, 5},3)_{simp}-polycycles, given in Appendix 1.

*(ii) Six* ({3, 4, 5},3)_{simp}-polycycles, infinite in one direction:

*(iii)* 21 = *infinite series obtained by taking two ends of the infinite polycycles from (ii) above and concatenating them*.

*For example, merging* α *with itself produces the infinite series of elementary* ({5},3)_{simp}-polycycles, denoted by E_{n} in [12]. See Figure 14.1 for the first 3 *members (starting with 6 faces) of two such series*: αα *and* βε.

*(iv) The infinite series of Barrel*_{m}, 2 ≤ *m* ≤ ∞, *and its nonorientable quotient for m odd*.

*Proof*. Take an elementary ({2, 3, 4, 5},3)_{simp}-polycycle *P* that, by Theorem 14.2, is kernel-elementary. If its kernel is empty, ...