Abstract: We first introduce a discrete quantum walk with Grover coin on an arbitrary finite graph whose links may vanish and reappear repeatedly during walker's evolution. We show that its asymptotic dynamics is dominated by so-called trapped states. These are pure states with a limited support on the underlying graph which stay undisturbed by evolution despite extremely hostile decoherence environment. It will be shown how graph properties determine a general structure of these trapped states and consequently how they impact an excitation transport along the graph to a desired sink. We briefly summarize its main features and discuss in more details the most counter-intuitive ones.