  In the graph sharing game, two players share a connected graph $G$ with
non-negative weights assigned to the vertices, claiming and collecting the
vertices of $G$ one by one, while keeping the set of all claimed vertices
connected through the whole game. Each player wants to maximize the total
weight of the vertices they have gathered by the end of the game, when the
whole $G$ has been claimed. It is proved that for any class $\mathcal{G}$ of
graphs with an odd number of vertices and with forbidden subdivision of a fixed
graph (e.g., for the class $\mathcal{G}$ of planar graphs with an odd number of
vertices), there is a constant $c_{\mathcal{G}}>0$ such that the first player
can secure at least the $c_{\mathcal{G}}$ proportion of the total weight of $G$
whenever $G\in\mathcal{G}$. Known examples show that such a constant does no
longer exist if any of the two conditions on the class $\mathcal{G}$ (an odd
number of vertices or a forbidden subdivision) is removed. The main ingredient
in the proof is a new structural result on weighted graphs with a forbidden
subdivision.
