Summary: Consider the following two-person game on the graph \(G\). Player I and II move alternatingly. Each move consists in coloring a yet uncolored vertex of \(G\) properly using a prespecified set of colors. The game ends when some player can no longer move. Player I wins if all of \(G\) is colored. Otherwise Player II wins. What is the minimal number \(\gamma(G)\) of colors such that Player I has a winning strategy? Improving a result of H. L. Bodlaender [Lect. Notes Comput. Sci. 484, 30-40 (1992; Zbl 0770.90098)] we show \(\gamma(G)\leq 4\) for each tree \(T\). We, furthermore, prove \(\gamma(G)= O(\log| G|)\) for graphs \(G\) that are unions of \(k\) trees. Thus, in particular, \(\gamma(G)= O(\log| G|)\) for the class of planar graphs. Finally we bound \(\gamma(G)\) by \(3\omega(G)- 2\) for interval graphs \(G\). The order of magnitude of \(\gamma(G)\) can generally not be improved for \(k\)-fold trees. The problem remains open for planar graphs.