Fig. 5

Network N^u and the simple graph resulting from the leaf connecting procedure. First, according to the pre-processing phase of the leaf connecting procedure, leaf 4 is deleted from the network because x is adjacent to two leaves. Then, x has degree 2 and thus needs to be suppressed. Then, first a pair of leaves is chosen and removed from the network, before the last leaf is removed (for (a), first leaves 1 and 2 are removed, followed by leaf 3; for (b), first leaves 1 and 3 are removed, followed by leaf 2 and for (c), first leaves 2 and 3 are removed, followed by leaf 1). Note that the graphs depicted in (a), (b) and (c) are isomorphic. Thus, here \( \mathcal{LCON}\left({N}^u\right) \) consists of a single simple graph (in general, \( \mathcal{LCON} \) can consist of several simple graphs; as an example see Fig. 6). Note, however, that the simple graph in \( \mathcal{LCON}\left({N}^u\right) \) differs from the simple graphs obtained from the leaf cutting and leaf shrinking procedures (cf. Fig. 2 and Fig. 4). Moreover, note that even though new vertices (a and b) were introduced, the total number of vertices of the simple graph in \( \mathcal{LCON}\left({N}^u\right) \) did not increase compared to N^u or even compared to N^u after the pre-processing step