Fig. 5

An illustration of how an input tree T _i is inserted into a network N _i−1 with the help of an embedded tree T ^′. a The network N _i−1 together with an embedded tree T ^′. b The input tree T _i , which will be embedded into N _i−1 by inserting the maximum acyclic agreement \(\mathcal {F}\) forest of T _i and T ^′ consisting of three components F _ρ , F ₁, and F ₂. c, d All important elements that have to be considered during the insertion of both components F ₁ and F ₂, respectively. Blue dots correspond to source nodes and red nodes to target nodes. Note that, regarding \(N_{i}^{(1)}\), there is only one valid pair of source and target nodes. Dashed edges are those edges that are disregarded when considering the restricted network in terms of T ^′ and the taxa set of the so far added components of \(\mathcal {F}\). e The resulting network N _i , which is obtained from \(N_{i}^{(3)}\) by suppressing each node of both in- and out-degree 1