A collection of splits Σ is compatible if and only if all splits are pairwise compatible.
Note that a trivial split of L is compatible with every split of L. Let Σ be a collection of splits of a label set X that includes all trivial splits.
Two different splits S1 = A1| B1, and S2 = A2| B2, are compatible, if one of the following conditions holds: A 1 ⊆ A 2, A 1 ⊆ B 2, B 1 ⊆ A 2 o r B 1 ⊆ B 2. A collection of splits is called compatible, if all possible pairing of splits are compatible.
Formally, a collection of splits of X is circular if there exists an ordering x1,⋯, x n of X such that every split is of the form { x i, x i +1,⋯, x j}| X - { x i,⋯,x j} for some i and j, 1 ≤ i ≤ j ≤ n.
Instead of something like "Anthony Roth Costanzo Baroque Ariass," his first album, thankfully, will be a collection of tracks split evenly between Handel and Philip Glass.
First, a collection of weighted splits is constructed from a distance matrix, and then these splits are represented using a splits graph, where a totally compatible collection of splits would be precisely represented as a tree, but incompatible splits as cycles or boxes.
A phylogenetic network can be constructed from a collection of weighted splits.
N-Net uses circular ordering to construct a collection of weighted splits.
The N-Net works in a similar way: it first produces a circular ordering from distance matrix and then constructs a collection of weighted splits.
The N-Net produces a circular ordering from a distance matrix, then constructs a collection of weighted splits using circular ordering.
