  A rotor walk in a directed graph can be thought of as a deterministic version
of a Markov Chain, where a pebble moves from vertex to vertex following a
simple rule until a terminal vertex, or sink, is reached. The ARRIVAL problem,
as defined by Dohrau and al., consists in determining which sink will be
reached. While the walk itself can take an exponential number of steps, this
problem belongs to the complexity class NP$\cap$co-NP without being known to be
in P. Several variants have been studied where we add one or two players to the
model, defining deterministic analogs of stochastic models (e.g., Markovian
decision processes, Stochastic Games) with rotor-routing rules instead of
random transitions. The corresponding decision problem address the existence of
strategies for players that ensure some condition on the reached sink. These
problems are known to be $NP$-complete for one player and $PSPACE$-complete for
two players. In this work, we define a class of directed graphs, namely
tree-like multigraphs, which are multigraphs having the global shape of an
undirected tree. We prove that the different variants of the reachability
problem with zero, one, or two players can be solved in linear time, while the
number of steps of rotor walks can still be exponential. To achieve this, we
define a notion of return flow, which counts the number of times the pebble
will bounce back in subtrees of the graph.
