On the oriented perfect path double cover conjecture

‎An  oriented perfect path double cover (OPPDC) of a‎ ‎graph $G$ is a collection of directed paths in the symmetric‎ ‎orientation $G_s$ of‎ ‎$G$ such that‎ ‎each arc‎ ‎of $G_s$ lies in exactly one of the paths and each‎ ‎vertex of $G$ appears just once as a beginning and just once as an‎ ‎end of a path‎. ‎Maxov{'a} and Ne{v{s}}et{v{r}}il (Discrete‎ ‎Math‎. ‎276 (2004) 287-294) conjectured that every graph except‎ ‎two complete graphs $K_3$ and $K_5$ has an   OPPDC and they‎ ‎claimed that the minimum degree of the minimal counterexample to‎ ‎this conjecture is at least four‎. ‎In the proof of their claim‎, ‎when a graph is smaller than the minimal counterexample‎, ‎they missed to consider the special cases $K_3$ and $K_5$‎. ‎In this paper‎, ‎among some‎ ‎other results‎, ‎we present the complete proof for this fact‎. ‎Moreover‎, ‎we prove that the minimal counterexample to this‎ ‎conjecture is $2$-connected and $3$-edge-connected‎.