Exact(3)
Clearly disconnecting is exactly two components: (336).
disconnects in exactly two components, so we can find an such that.
It is obvious that (G-P_{v_{1}v_{2}cdots v_{x}}) has exactly two components, say (G_{1}) and (G_{2}), both without pendant vertices and with (c(G_{i})< c(G)), (i=1,2).
Similar(55)
(This terminology comes from the obsolete Kerberos version 4, where a principal name had exactly three components: name, instance and realm). A Kerberos principal name having only one component is sometimes called a "null instance", for the same historical reasons.
Deletion of all edges corresponding to a split divides the network in exactly two connected components, one containing all taxa from one part of the split, and another containing taxa from the other.
It is well-known (see, e.g., [11]) that has exactly two connected components.
Assume that G (P ) − Ψ (e ) has exactly two connected components.
If F is a minimal separator of LG (P ), then LG (P ) − F has exactly two connected components.
Thus, if G (P ) − Ψ (e ) has exactly two connected components, Ψ e) is a minimal cut of G (P ).
Then, for every edge e∈ E(S), (i) Ψ e) is a cut of G (P ) and (ii) Ψ e) is a minimal cut of G (P ) if and only if G (P ) − Ψ (e ) has exactly two connected components.
This is an unrooted phylogenetic network with the property that every split S= A∣ B in Σ is represented in N by a set of parallel edges, such that deleting all edges in the set will result in exactly two connected components, one labeled by the taxa in A and the other labeled by B. A class of splits of particular interest are the circular splits.
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