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Q- Can someone explain the equivalence in detail please?

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The correspondence is almost the same as your other question. Here, the starting node is specified, so you don't need a dummy node or dummy arcs out of the starting node. To go from H-path (with start $1$ and end $t$) to tour, insert arc $(t,1)$. To go from tour to H-path, remove the arc that enters $1$.

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