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

For 2 ≤ *m* ≤ ∞, call *snub m-antiprism* the ({3},5)_{simp}-polycycle with two *m*-gonal holes separated by 4*m* 3-gonal proper faces. This polycycle is nonextensible if and only if *m* ≤ 4. Its symmetry group, *D*_{md}, coincides with the symmetry group of an underlying 5-valent plane graph if and only if *m* =3. See below for pictures of a snub *m*-antiprism for *m* = 2, 3, 4, 5, and ∞ (see [4], p. 119 for a formal definition):

**Theorem 14.5** *The list of elementary* ({2, 3},5)*-polycycles consists of*:

*(i) 57 sporadic* ({2, 3},5)_{simp}-polycycles, given in Appendix 2.

*(ii) The following* 3 *infinite* ({2, 3},5)_{simp}-polycycles:

*(iii)* 6 *infinite series of* ({2, 3},5)_{simp}-polycycles with one hole (obtained by concatenating endings of a pair of polycycles, given in (ii); see Figure 14.2 for the first 4 *polycycles)*.

*(iv) The infinite series of snub m-antiprisms for* 2 ≤ *m* ≤ ∞ *and its nonorientable quotient for m odd*.

*Proof*. Take an elementary ({2, 3},5)_{simp}-polycycle *P* that, by Proposition 14.2, is kernel-elementary. If its kernel is empty, then *P* is simply an *r*-gon with *r* ∈{2, 3}. If the kernel is reduced to a vertex, then *P* is simply a 5-tuple of 2-, 3-gons. If three edges *e*_{i} = {*v*_{i–1}, *v*_{i}}, *i* =1,..., 3 are part of the kernel with *e*_{1}, *e*_{2} and *e*_{2}, *e*_{3} not part of an ...