We study the following combinatorial game played by two players, Alice and
Bob, which generalizes the Pizza game considered by Brown, Winkler and others.
Given a connected graph G with nonnegative weights assigned to its vertices,
the players alternately take one vertex of G in each turn. The first turn is
Alice's. The vertices are to be taken according to one (or both) of the
following two rules: (T) the subgraph of G induced by the taken vertices is
connected during the whole game, (R) the subgraph of G induced by the remaining
vertices is connected during the whole game.