On objects dual to tree-cut decompositions

Abstract

Tree-cut width is a graph parameter introduced by Wollan [Wol15] that is an analogue of treewidth for the immersion order on graphs in the following sense: the tree-cut width of a graph is functionally equivalent to the largest size of a wall that can be found in it as an immersion. In this work we propose a variant of the definition of tree-cut width that is functionally equivalent to the original one, but for which we can state and prove a tight duality theorem relating it to naturally de ned dual objects: appropriately de ned brambles and tangles. Using this result we also propose a game characterization of tree-cut width. * This work is a part of projects CUTACOMBS (ŁB, OD, KO) and TOTAL (MP) that have received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreements No. 714704 and No. 677651, respectively).