  Tree-cut width is a parameter that has been introduced as an attempt to
obtain an analogue of treewidth for edge cuts. Unfortunately, in spite of its
desirable structural properties, it turned out that tree-cut width falls short
as an edge-cut based alternative to treewidth in algorithmic aspects. This has
led to the very recent introduction of a simple edge-based parameter called
edge-cut width [WG 2022], which has precisely the algorithmic applications one
would expect from an analogue of treewidth for edge cuts, but does not have the
desired structural properties. In this paper, we study a variant of tree-cut
width obtained by changing the threshold for so-called thin nodes in tree-cut
decompositions from 2 to 1. We show that this "slim tree-cut width" satisfies
all the requirements of an edge-cut based analogue of treewidth, both
structural and algorithmic, while being less restrictive than edge-cut width.
Our results also include an alternative characterization of slim tree-cut width
via an easy-to-use spanning-tree decomposition akin to the one used for
edge-cut width, a characterization of slim tree-cut width in terms of forbidden
immersions as well as approximation algorithm for computing the parameter.
