Vertex-split Octahedron

The vertex-split Octahedron and the vertex-split Icosahedron are polycycles obtained from Octahedron and Icosahedron, respectively, by splitting a vertex into two vertices and the edges, incident to it, into two parts, accordingly. The vertex-split Octahedron is drawn on Figure 4.2,1 and both of them are given on Figure 8.3 they are the only, besides five Platonic r, q) — f, non-extensible finite (r, )-polycycles. 

A natural parameter to measure an (r, 4)-helicene, will be the degree of the corresponding homomorphism into r, q (on vertices, edges, and faces). For q > 4, helicenes appear with vertices, but not edges, having same homomorphic image. The vertex-split Octahedron is a unique such maximal helicene for (r, q) = (3,4) (two 2-valent vertices are such see Figure 4.2). There is a finite number of such helicenes for (r, q) = (3, 5) one of them is the vertex-split Icosahedron. 

Four exceptional non-extensible polycycles, depicted on Figure 8.3 are vertex-split Octahedron, vertex-split Icosahedron, and two infinite ones Pi x Pi = Prism<, and Tr% = APrismoo (see Section 4.2). 

Lemma 8.23 ([DSS06]) All finite elliptic non-extensible (r, q)-polycycles are two vertex-splittings (of Octahedron and Icosahedron see first two in Figure 8.3) and five Platonic r, q) (with a face deleted). 

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