On-line Ramsey numbers for paths and stars

We study on-line version of size-Ramsey numbers of graphs defined via a game played between Builder and Painter: in one round Builder joins two vertices by an edge and Painter paints it red or blue. The goal of Builder is to force Painter to create a monochromatic copy of a fixed graph H in as few rounds as possible. The minimum,number of rounds (assuming both players play perfectly) is the on-line Ramsey number r̃(H) of the graph H. We determine exact values of r̃(H) for a few short paths and obtain a general upper bound r̃(P_n) ≤ 4n - 7. We also study asymmetric version of this parameter when one of the target graphs is a star S_n with n edges. We prove that r̃(S_n, H) ≤ n · e(H) when H is any tree, cycle or clique.