Fig. 14

Underlying pruned subgraphs and corresponding intersection graphs of a bubble with a single 6-cycle \(\mathcal {Y}\) (solid edges). Dotted edges are exclusive to paths and dashed gray edges are pruned out. In i and ii, \(\mathcal {Y}\) intersects with three valid 4-paths \(\textsf{A}\alpha \textsf{a}\), \(\textsf{B}\beta \textsf{b}\) and \(\textsf{C}\gamma \textsf{c}\). In i, the yellow solution including \(\mathcal {Y}\) would also include the three 2-paths \(\textsf{Ab}\), \(\textsf{Bc}\) and \(\textsf{Ca}\), being clearly superior. In ii, the yellow solution including \(\mathcal {Y}\) would still include the 2-path \(\textsf{Ba}\), having the same score of the green solution with three 4-paths. In any of the two cases, the underlying graph cannot be extended. In iii, \(\mathcal {Y}\) has plug connections with unsaturated path-lines starting at 4-paths \(\textsf{A}\alpha \textsf{a}\) and \(\textsf{B}\beta \textsf{b}\) (both can be extended)