Graph Sharing Game and the Structure of Weighted Graphs with a Forbidden Subdivision

The graph sharing game is played by two players, Alice and Bob, on a connected graph G with non-negative weights assigned to the vertices. Starting with Alice, the players take the vertices of G one by one, in each move keeping the set of all taken vertices connected, until the whole G has been taken. Each player wants to maximize the total weight of the vertices they have gathered. It is proved that for any class G of graphs with an odd number of vertices and with forbidden subdivision of a fixed graph, there is a constant cG > 0 such that Alice can guarantee herself at least cG of the total weight of G whenever G ∈ G. Known examples show that this is no longer true if any of the two conditions (odd number of vertices or a forbidden subdivision) on the class G is dropped. The main ingredient in the proof is a structural result on weighted graphs with a forbidden subdivision, which may be of independent interest.