Reduced Partial 3-Tree Is Unique

Claim/Theorem

Let \(M\) be a 3-connected matroid, and let \(T\) be a partial 3-tree displaying the nontrivial 3-separations of \(M\) up to the natural equivalence from [[maximal-partial-3-tree-displays-all-nonsequential-3-separations.md]].

Impose the source paper's natural reduction rule:

from every pair of edges that meet at a degree-2 vertex and whose other ends both have degree at least 3, contract one of those two edges.

Then the resulting reduced partial 3-tree is unique.

Equivalently, the nonsequential cut-rank-2 template of a 3-connected matroid has a canonical reduced form. So once one is in the partial-3-tree regime, there is no longer arbitrary decomposition choice.

Dependencies

• [[maximal-partial-3-tree-displays-all-nonsequential-3-separations.md]]

Conflicts/Gaps

• This theorem still lives entirely in the exact 3-separation regime, i.e. intrinsic cut rank 2.
• Uniqueness of the reduced tree does not itself exclude that template from occurring in Quantum Tanner parity-check matroids.
• It also says nothing about sequential 3-separations, which remain a distinct low-order loophole.

Sources

• 10.1016/j.ejc.2006.01.007