Form the product graph $\Gamma \times \Gamma$ whose vertices are pairs $(u, u') \in \Gamma \times \Gamma$ and paths $(p, p') : (u, u') \to (v, v')$ are pairs of paths $p : u \to v$, $p' : u' \to v'$. Let $\Delta = \lbrace (u, u) \mid u \in V(\Gamma) \rbrace$ be the diagonal. Your question is equivalent to the question whether $\Delta$ can be reached from a given vertex $(x,y)$ in the graph $\Gamma \times \Gamma$ (and finding such a vertex, but most algorithms automatically give you one). This does not seem to be that hard to solve as it is just a reachability problem. If you want it in classical form, attach every vertex in $\Delta$ to a special vertex $\infty$ and ask for a path from $(x,y)$ to $\infty$. There are algorithms which preprocess the graphs so that subsequent reachability queries can be answered fast, in case you have to do this a lot (Wikipedia explains some of this).

Form the product graph $\Gamma \times \Gamma$ whose vertices are pairs $(u, u') \in \Gamma \times \Gamma$ and edges $(p, p') : (u, u') \to (v, v')$ are pairs of edges $p : u \to v$, $p' : u' \to v'$. Let $\Delta = \lbrace (u, u) \mid u \in V(\Gamma) \rbrace$ be the diagonal. Your question is equivalent to the question whether $\Delta$ can be reached from a given vertex $(x,y)$ in the graph $\Gamma \times \Gamma$ (and finding such a vertex, but most algorithms automatically give you one). This does not seem to be that hard to solve as it is just a reachability problem. If you want it in classical form, attach every vertex in $\Delta$ to a special vertex $\infty$ and ask for a path from $(x,y)$ to $\infty$. There are algorithms which preprocess the graphs so that subsequent reachability queries can be answered fast, in case you have to do this a lot (Wikipedia explains some of this).

Form the product graph $\Gamma \times \Gamma$ whose vertices are pairs $(u, u') \in \Gamma \times \Gamma$ and paths $(p, p') : (u, u') \to (v, v')$ are pairs of paths $p : u \to v$, $p' : u' \to v'$. Let $\Delta = \lbrace (u, u) \mid u \in V(\Gamma) \rbrace$ be the diagonal. The question is then whether $\Delta$ can be reached from a given vertex $(x,y)$ in $\Gamma \times \Gamma$. This does not seem to be that hard to solve as it is just a reachability problem (if you want it in classical form, attach every vertex in $\Delta$ to a special vertex $\infty$ and ask for a path from $(x,y)$ to $\infty$).

Form the product graph $\Gamma \times \Gamma$ whose vertices are pairs $(u, u') \in \Gamma \times \Gamma$ and paths $(p, p') : (u, u') \to (v, v')$ are pairs of paths $p : u \to v$, $p' : u' \to v'$. Let $\Delta = \lbrace (u, u) \mid u \in V(\Gamma) \rbrace$ be the diagonal. Your question is equivalent to the question whether $\Delta$ can be reached from a given vertex $(x,y)$ in the graph $\Gamma \times \Gamma$ (and finding such a vertex, but most algorithms automatically give you one). This does not seem to be that hard to solve as it is just a reachability problem. If you want it in classical form, attach every vertex in $\Delta$ to a special vertex $\infty$ and ask for a path from $(x,y)$ to $\infty$. There are algorithms which preprocess the graphs so that subsequent reachability queries can be answered fast, in case you have to do this a lot (Wikipedia explains some of this).