  We present a generalized definition of discrete-time quantum walks convenient
for capturing a rather broad spectrum of walker's behavior on arbitrary graphs.
It includes and covers both: the geometry of possible walker's positions with
interconnecting links and the prescribed rule in which directions the walker
will move at each vertex. While the former allows for the analysis of
inhomogeneous quantum walks on graphs with vertices of varying degree, the
latter offers us to choose, investigate, and compare quantum walks with
different shift operators. The synthesis of both key ingredients constitutes a
well-suited playground for analyzing percolated quantum walks on a quite
general class of graphs. Analytical treatment of the asymptotic behavior of
percolated quantum walks is presented and worked out in details for the Grover
walk on graphs with maximal degree 3. We find, that for these walks with cyclic
shift operators the existence of an edge-3-coloring of the graph allows for
non-stationary asymptotic behavior of the walk. For different shift operators
the general structure of localized attractors is investigated, which determines
the overall efficiency of a source-to-sink quantum transport across a
dynamically changing medium. As a simple nontrivial example of the theory we
treat a single excitation transport on a percolated cube.
