Fig. 1

In a we have the cp-graph \(G^{X}_{\pi }\) for \(\pi = (+\,4 +2 +3 -1 -5)\) and \(X = (3, 0, 0, {-3}, 0)\), with \(cn(X) = 5\), \(cc(G^{X}_{\pi }) = 2\), and \(cc^-(G^{X}_{\pi }) = 2\). In b we have the cp-graph \(G^{X'}_{\pi }\) for \(X' = T_{1,4}(X) = (-\,2, 0, 0, 2, 0)\). By Property 1 \(cn(X') = cn(X) + 2(n - x_1 + x_4) = 5-2 = 3\) (recall that \(cn(X')\) is also the sum of weights in the graph \(G^{X'}_{\pi }\)), \(cc(G^{X'}_{\pi }) = 3\), and \(cc^-(G^{X'}_{\pi }) = 0\). We can see in \(G^{X}_{\pi }\) and \(G^{X'}_{\pi }\) that the 2-reversal \(\rho (4,5)\) is induced by \(X'\) but not by X; c the cp-graph \(G^{X''}_{\pi '}\) for \(\pi ' = \pi \cdot \rho (4,5) = (+\,4 +2 +3 +5 +1)\) and its VD-vector \(X'' = ({-2}, 0, 0, 1, 1)\) with \(cn(X'') = cn(X')-1 = 2\), \(cc(G^{X''}_{\pi '}) = 3\), and \(cc^-(G^{X''}_{\pi '}) = 0\)