Games for width parameters and monotonicity
Department of Informatics, University of Bergen, N-5020 Bergen, Norway
Abstract 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.