Fig. 3

a–f The six possible configurations for VD-vector \(X \in S\), which we call \(X^a, X^b, X^c, X^d, X^e, X^f\). For VD-vectors \(X^a\) and \(X^f\), the union of intervals, highlighted in gray, contains all the elements, thus \(cc(G^{X^a}_{\pi }) = cc(G^{X^f}_{\pi }) = 1\). For VD-vectors \(X^b\) to \(X^e\), the union of intervals does not necessarily contain all elements (cf. white regions), but for each of these configurations there exists a VD-vector \(X' \in S'\) obtained by transforming X, shown in g–j, in which the union of intervals, highlighted in gray, also contains all elements, i.e., \(cc(G^{X'}_{\pi }) = 1\). Note that in g \(X' = T_{k,j}(X^b)\), in h \(X' = T_{i,j}(X^c)\), in i \(X' = T_{k,j}(X^d)\), and in j \(X' = T_{k,l}(X^e)\)