1. 1. Is it true that, for all G and k, white cannot win the game?

2. 2. Does there exist a tree T and a positive integer k for which the outcome is a draw?

Let τ(G) denote the covering number of G, i.e., τ(G) is the minimum number of vertices covering all edges of G. We prove that black wins the game if τ(G) k. We use this result to show that white never wins the game if Gis bipartite, thus providing a partial answer to the first question. We answer the second question in the affirmative by constructing an infinite series of trees for which the outcome is a draw (for some k). Moreover, we present results on extremal problems arising in the context of the game. We also completely solve the cases when G is a path or a cycle. Further, we completely settle the case k 2. In the proofs of our results, matchings and cycles in graphs play a predominant role.