  Quantum walks exhibit properties without classical analogues. One of those is
the phenomenon of asymptotic trapping -- there can be non-zero probability of
the quantum walker being localised in a finite part of the underlying graph
indefinitely even though locally all directions of movement are assigned
non-zero amplitudes at each step. We study quantum walks with the flip-flop
shift operator and the Grover coin, where this effect has been identified
previously. For the version of the walk further modified by a random dynamical
disruption of the graph (percolated quantum walks) we provide a recipe for the
construction of a complete basis of the subspace of trapped states allowing to
determine the asymptotic probability of trapping for arbitrary finite connected
simple graphs, thus significantly generalizing the previously known result
restricted to planar 3-regular graphs. We show how the position of the source
and sink together with the graph geometry and its modifications affect the
excitation transport. This gives us a deep insight into processes where
elongation or addition of dead-end subgraphs may surprisingly result in
enhanced transport and we design graphs exhibiting this pronounced behavior. In
some cases this even provides closed-form formulas for the asymptotic transport
probability in dependence on some structure parameters of the graphs.
