  We introduce a search game for two players played on a "scenario" consisting
of a ground set together with a collection of feasible partitions. This general
setting allows us to obtain new characterisations of many width parameters such
as rank-width and carving-width of graphs, matroid tree-width and
GF(4)-rank-width. We show that the monotone game variant corresponds to a tree
decomposition of the ground set along feasible partitions. Our framework also
captures many other decompositions into "simple" subsets of the ground set,
such as decompositions into planar subgraphs.
  Within our general framework, we take a step towards characterising monotone
search games. We exhibit a large class of "monotone" scenarios, i.e. of
scenarios where the game and its monotone variant coincide. As a consequence,
determining the winner is in NP for these games. This result implies
monotonicity for all our search games, that are equivalent to branch-width of a
submodular function.
  Finally, we include a proof showing that the matroid tree-width of a graphic
matroid is not larger than the tree-width of the corresponding graph. This
proof is considerably shorter than the original proof and it is purely graph
theoretic.
