=. ?. , Assume that f (v 2 ) = j for some j ? {?, 1}. Start by switching some of v 5

=. ?. Now, choose some i ? N ?(v 4 v 8 ) (f (v 4 )) \ {?, j} and set f (v 8 ) = i. According to Observation 6.5, there is an sp-homomorphism g of the signed path induced by the vertices v 2

?. {f, Up to the left/right symmetry, we may also assume that f (v 4 ) ? {f (v 1 ), f (v 2 )}, i.e. {f (v 1 ), f (v 2 )} = {f (v 3 ), f (v 4 )}. We here consider two subcases: (a) f (v 1 ) = f (v 3 ) and f

, Without loss of generality, assume f (v 1 ) = f (v 3 ) = ? and f (v 2 ) = f (v 4 ) = 1. Start by switching v 5

, Now choose i ? N ?(v 2 v 6 ) (1)\ {?}, j ? N ?(v 4 v 8 ) (1) \ {?, i} and set f (v 6 ) = i and f (v 8 ) = j

, if needed) to make sure that ?(v 2 v 6 ) = + and ?(v 4 v 8 ) = ?. Up to exchanging (v 1 , v 5 ) with (v 3 , v 7 ), we may assume that ?(v 5 v 6 ) = ?(v 5 v 8 ) and ?(v 6 v 7 ) = ?, ?, then set f, vol.3

, Without loss of generality, assume f (v 1 ) = f (v 4 ) = ? and f (v 2 ) = f (v 3 ) = 1. Start by switching some of v 5

, if ?(v 6 v 7 ) = +, and f (v 7 ) = 4 otherwise, vol.2

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