A generalized cover time for random walks on graphs

Given a random walk on a graph, the cover time is the first time (number of steps) that every vertex has been hit (covered) by the walk. Define the marking time for the walk as follows. When the walk reaches vertex v(i), a coin is flipped and with probability pi the vertex is marked (or colored). We study the time that every vertex is marked. (When all the pi's are equal to 1, this gives the usual cover time problem.) General formulas are given for the marking time of a graph. Connections are made with the generalized coupon collector's problem. Asymptotics for small p(i)'s are given. Techniques used include combinatorics of random walks, theory of determinants, analysis and probabilistic considerations.