## 14.6 Conclusion

The classification, developed here, is an extension of previous work of Deza and Shtogrin. It allows for various possible numbers of sides of interior faces and several possible holes. A natural question is if one can further enlarge the class of polycycles.

**Figure 14.2** The first 4 members of the six infinite series of ({2, 3},5)_{simp}-polycycles from Theorem 14.5(iii).

There will be only some technical difficulties if one tries to obtain the catalog of elementary (*R*, *Q*)*-polycycles*, that is, the generalization of the (*R*, *q*) polycycle allowing the set *Q* for values of a degree of interior vertices. Such a polycycle is called an *elliptic*, a *parabolic*, or a *hyperbolic* if (where *r* = *max*_{i ∈}_{R}^{i}, *q* = *max*_{i ∈}_{Q}^{i}) is positive, zero, or negative, respectively. The decomposition and other main notions could be applied directly.

We required 2-connectivity and that any two holes not share a vertex. If one removes those two conditions, then too many other graphs appear.

The omitted cases (*R*, *q*) = ({2}, *q*) are not interesting. In fact, consider the infinite series of ({2}, 6)-polycycles, *m-bracelets*, *m* ≥ 2(i.e., *m*-circle with each edge being tripled). The central edge is a bridge for those ...