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Figure 3 | Algorithms for Molecular Biology

Figure 3

From: Constructing perfect phylogenies and proper triangulations for three-state characters

Figure 3

Minimal separators and proper clusters. In this figure, the bipartition ab|cdef gives rise to the proper clusters ab and cdef. The shared character states χ 1 2 , χ 0 3 , χ 0 4 form a legal minimal separator S in G(M). G(M) − S has three connected components, of which two are full (components C1 and C2). The S-partition gives rise to the bipartition because t(C1) = {a, b} and t(C2) t(C3) = {c, d, e, f}.T is a clique tree for G(M) (in this case, G(M) happens to be chordal). T is obtained fromT by resolving the nodes labeled b, c, f. Note that S is represented inT on edge bc because { χ 0 1 , χ 1 2 , χ 0 3 , χ 0 4 } { χ 0 1 , χ 1 2 , χ 0 3 , χ 0 4 } = { χ 1 2 , χ 0 3 , χ 0 4 } . For a clique treeT of a chordal graph, every minimal separator of the chordal graph behaves this way[11, 21]. In this sense, legal minimal separators are analagous to splitting vectors.

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