**Theorem 14.4** *Any elementary* ({2, 3},4)*-polycycle is one of the following eight*:

*Proof*. The list of elementary ({3},4)_{simp}-polycycles is determined in [3] and consists of the first four graphs of this theorem. Let *P* be a ({2, 3}, 4)-polycycle containing a 2-gon. If|*F*_{1}| = 1, then it is the 2-gon. Clearly, the case where two 2-gons share one edge is impossible. Assume that *P* contains two 2-gons that share a vertex. Then we should add a triangle on both sides and thus obtain the second polycycle given above. If there is a 2-gon that does not share a vertex with a 2-gon, then *P* contains the following pattern:

Thus, clearly, *P* is one of the last two possibilities above.

Note that the seventh and fourth polycycles in Theorem 14.4 are, respectively, 2- and 3-antiprisms; here the exterior face is the unique hole. The *m*-antiprism for any *m* ≥ 2 can also be seen as a ({2, 3}, 4)-polycycle with *F*_{2} consisting of the exterior and interior *m*-gons; this polycycle is not elementary.

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