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Using this structural result, we show that every finite bridgeless $(P/e)$-minor-free multigraph admits a nowhere-zero $4$-flow. This extends the theorem of Wang, Zhang and Zhang (2009) for graphs excluding the graph obtained by contracting three edges of a perfect matching of $P$, and complements the theorem of Thomas and Thomson (2000) for $(P-e)$-minor-free graphs. 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