Canonical sequence conjecture for tangle-distinguishing tree-decompositions

Let $G$ be a graph, and let $n$ denote the relevant maximum tangle order. A canonical sequence of tree-decompositions is a sequence $(T_i,\mathcal{V}_i)$ for $1\leq i\leq n$, where each $(T_i,\mathcal{V}_i)$ is a tree-decomposition of $G$, and where refinement and torsos are understood in the usual sense. A part is inessential at stage $i$ if it does not contain an $(i+1)$-tangle.

Canonical sequence conjecture. For every graph $G$ there exists a canonical sequence of tree-decompositions $(T_i,\mathcal{V}_i)$ for $1 \leq i \leq n$ of $G$ such that:

• $(T_i,\mathcal{V}_i)$ distinguishes every $i$-tangle in $G$ for each $i$;
• $(T_n,\mathcal{V}_n)$ distinguishes the set of maximal tangles in $G$;
• $(T_{i+1},\mathcal{V}_{i+1})$ refines $(T_i,\mathcal{V}_i)$ for each $i$;
• the torso of every inessential part in $(T_i,\mathcal{V}_i)$ has branch-width $<i$.

This is presented as the strongest form of a positive answer to the preceding question about bounding the branch-width of inessential parts. The preceding theorem establishes the existence of a canonical nested set distinguishing all tangles, but the claimed simultaneous sequence, refinement property, and branch-width bound remain conjectural.