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${\chi }_1^2,{\chi }_0^3,{\chi }_0^4$ 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}.$\mathcal{T}$ is a clique tree for G(M) (in this case, G(M) happens to be chordal). T is obtained from$\mathcal{T}$ by resolving the nodes labeled b, c, f. Note that S is represented in$\mathcal{T}$ on edge bc because$\{{\chi }_0^1,{\chi }_1^2,{\chi }_0^3,{\chi }_0^4\}\cap \{{\chi }_0^1,{\chi }_1^2,{\chi }_0^3,{\chi }_0^4\}=\{{\chi }_1^2,{\chi }_0^3,{\chi }_0^4\}$. For a clique tree$\mathcal{T}$ 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.