Fig. 6

Illustration of the co-optimality of every valid 4-cycle not intersecting a 2-cycle in the \(\sigma _4\) disambiguation. In each of these pictures, each gray path is necessarily odd with length at least one and alternates \(\check{\mathbb {D}}\)- and \(\mathbb {S}\)-edges. Furthermore, the 4-cycle \(C=(uvwz)\) is displayed in the center, induced by blue edges. In (i) it is easy to see that any optimal solution is induced by the blue edges and includes, besides the cycle C, cycles \((\hat{u}\ldots \hat{v})\) and \((\hat{w}\ldots \hat{z})\). In ii an optimal solution includes 4-cycle C and cycle \(C'=(\hat{u}\hat{v}\ldots \hat{w}\hat{z}\ldots )\). If the connection between \(\hat{v}\) and \(\hat{w}\) is a single edge, then another optimal solution is induced by the red edges, including 4-cycle \(D=(u\hat{v}\hat{w}z)\) and cycle \(D'=(v\hat{u}\ldots \hat{z}w)\). And if additionally the connection between \(\hat{u}\) and \(\hat{z}\) is a single edge, then both \(C'\) and \(D'\) are also 4-cycles. In (iii) any optimal solution is induced by the blue edges and includes 4-cycle C and cycle \((\hat{u}\hat{v}\ldots \hat{z}\hat{w}\ldots )\), which is also a 4-cycle when the connections between \(\hat{v}\) and \(\hat{z}\) and between \(\hat{u}\) and \(\hat{w}\) are single edges