Why do we need a middle vertex in reducing a DHP to UHP?

I want to reduce a Direct Hamilton Path(DHP) to an Undirected Hamilton Path(UHP), and for that matter the standard algorithm is to split each vertex say v in DHP into 3 vertices namely vIN, vMID and vOUT. My question is that why do we need the middle vertex i.e. vMID? All these vertices are anyways connected with each other. enter image description here



Read more here: https://stackoverflow.com/questions/67378912/why-do-we-need-a-middle-vertex-in-reducing-a-dhp-to-uhp

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