Fig. 4

(i) The gray path connecting vertex \(\hat{v}\) to vertex \(\hat{u}\) is necessarily odd with length at least one and alternates \(\check{\mathbb {D}}\)- and \(\mathbb {S}\)-edges. The 2-cycle \(C=(uv)\) intersects the longer cycle \(D=(u\hat{v}\ldots \hat{u}v)\). Any solution containing (red edges) \(\widetilde{\mathcal {E}}=\{u\hat{v}, \hat{u}v\}\) induces D and can be improved by switching \(\widetilde{\mathcal {E}}\) to (blue edges) \(\mathcal {E}=\{uv, \hat{u}\hat{v}\}\), inducing, instead of D, the 2-cycle C and cycle \(D'=(\hat{v}\ldots \hat{u})\) (which is shorter than D). (ii) The gray paths connecting vertex \(\hat{v}\) to telomere y and vertex \(\hat{u}\) to telomere z alternate \(\check{\mathbb {D}}\)- and \(\mathbb {S}\)-edges. The 2-cycle \(C=(uv)\) intersects the longer path \(P=y\ldots \hat{v}uv\hat{u}\ldots z\). Any solution containing (red edges) \(\widetilde{\mathcal {E}}=\{u\hat{v}, \hat{u}v\}\) induces P and can be improved by switching \(\widetilde{\mathcal {E}}\) to (blue edges) \(\mathcal {E}=\{uv, \hat{u}\hat{v}\}\), inducing, instead of P, the 2-cycle C and path \(P'=y\ldots \hat{v}\hat{u}\ldots z\) (which is of the same type, but 2 edges shorter than P)