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Lean tree-cut decompositions: obstructions and algorithms

The notion of tree-cut width has been introduced by Wollan [The structure of graphs not admitting a fixed immersion. Journal of Combinatorial Theory, Series B, 110:47 - 66, 2015]. It is defined via tree-cut decompositions, which are tree-like decompositions that highlight small (edge) cuts in a graph. In that sense, tree-cut decompositions can be seen as an edge-version of tree-decompositions and have algorithmic applications on problems that remain intractable on graphs of bounded treewidth. In this paper, we prove that every graph admits an optimal tree-cut decomposition that satisfies a certain Menger-like condition similar to that of the lean tree decompositions of Thomas [A menger-like property of tree-width: The finite case. Journal of Combinatorial Theory, Series B, 48(1):67 - 76, 1990]. This allows us to give, for every k∈N, an upper-bound on the number immersion-minimal graphs of tree-cut width k. Our results imply the constructive existence, for every fixed r, of a linear algorithm for deciding whether the tree-cut width of a graph is at most r.

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