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Hence Δ G ′ (M ) ⊆ Δ G (M ) L. Conversely, let S ∈ Δ G (M ) L. First we show that if no F-edge f of G ′ (M) crosses S (i.e. f =  xy where S separates x and y), then S is a minimal separator of G ′ (M).

Each minimal separator of G ′ (M) is legal by Theorem 3.1.

Let F be any minimal separator of LG (P ) and u be any vertex of any input tree.

If F is a minimal separator of LG (P ), then LG (P ) − F has exactly two connected components.

Hence S ′ is illegal, and any legal minimal separator of G(M) is not crossed by any F-edge.

Then, Inc u)⊆ F. But F is a minimal separator of LG (P ), and by Lemma 3, Inc u) ⊈ F, a contradiction.

Let Δ G denote the minimal separators of graph G.

Then the legal minimal separators of G M) are exactly the minimal separators of G ′ (M) (i.e., Δ G ′ (M ) = Δ G (M ) L ).

□ Let Δ G (M ) L denote the set of legal minimal separators of G M).

□ This suggests that analyzing the minimal separators of G ′ (M) suffices for 3-state construction.

This proves our characterization of parallel minimal separators of Δ G (M ) ∗.