Fig. 16

While a cycle-line can be arbitrarily large, by increasing the complexity of a bubble we quickly saturate the space for adding cycles to it. Starting with (a) a simple cycle-line of length two, we can either (b1) connect the open vertices of squares 2 and 3, obtaining a cyclic cycle-line of length 4 that cannot be extended, or (b2) extend the line so that it achieves length three. From (b2) we can obtain (c1) a cyclic cycle-line of length 4 that can be extended first by adding cycle \(\mathcal {Y}_5\) next to \(\mathcal {Y}_1\) and then either adding \(\mathcal {Y}_5'\) next to \(\mathcal {Y}_3\) or closing \(\mathcal {Y}_6\), \(\mathcal {Y}_7\) and \(\mathcal {Y}_8\) so that we get (c2). In both cases no further extensions are possible. Note that (c2) can also be obtained by extending a cycle-line of length three and transforming it into a star with three branches, that can still be extended by closing \(\mathcal {Y}_3\), \(\mathcal {Y}_6\), \(\mathcal {Y}_7\) and \(\mathcal {Y}_8\). (These steps are more elaborated in of Additional file 1: Figs. S1–S6)