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Analysis of Complex Networks by Frank Emmert-Streib, Matthias Dehmer

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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.

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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 ifimages (where r = maxiRi, q = maxiQi) 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 ...

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