On objects dual to tree-cut decompositions

TL;DR

This paper introduces a variant of tree-cut width, establishing a duality theorem with brambles and tangles, and provides a game-based characterization, enhancing understanding of this graph parameter.

Contribution

It proposes a new variant of tree-cut width with a proven duality theorem linking it to brambles and tangles, and introduces a game characterization.

Findings

Established a tight duality theorem for the new tree-cut width variant.

Defined dual objects: brambles and tangles, for the new parameter.

Presented a game characterization of tree-cut width.

Abstract

Tree-cut width is a graph parameter introduced by Wollan 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 defined dual objects: appropriately defined brambles and tangles. Using this result we also propose a game characterization of tree-cut width.