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We restrict our study to cubic graphs, because for each non-cubic signed graph $(G,\\sigma)$ there is a set ${\\cal G}(G,\\sigma)$ of cubic graphs such that $F(G, \\sigma) \\leq \\min \\{F(H,\\sigma_H) : (H,\\sigma_H) \\in {\\cal G}(G)\\}$. We prove that $F(G,\\sigma) \\leq 6$ if $(G,\\sigma)$ contains a bridge and $F(G,\\sigma) \\leq 7$ in general. We prove better bounds, if there is an element $(H,\\sigma_H)$ of ${\\cal G}(G,\\sigma)$ which satisfies some additional conditions. 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We restrict our study to cubic graphs, because for each non-cubic signed graph $(G,\\sigma)$ there is a set ${\\cal G}(G,\\sigma)$ of cubic graphs such that $F(G, \\sigma) \\leq \\min \\{F(H,\\sigma_H) : (H,\\sigma_H) \\in {\\cal G}(G)\\}$. We prove that $F(G,\\sigma) \\leq 6$ if $(G,\\sigma)$ contains a bridge and $F(G,\\sigma) \\leq 7$ in general. We prove better bounds, if there is an element $(H,\\sigma_H)$ of ${\\cal G}(G,\\sigma)$ which satisfies some additional conditions. 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