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Maximal Partial 3-Tree Displays All Nonsequential 3-Separations

Claim/Theorem

Let \(M\) be a 3-connected matroid with

\[ |E(M)|\ge 9. \]

Then every maximal partial 3-tree for \(M\) displays, up to the paper's natural equivalence relation, every nonsequential 3-separation of \(M\).

Equivalently, the entire cut-rank-2 regime is already highly structured in 3-connected matroids: once sequential 3-separations are ignored, all remaining exact 3-separations can be organized by one tree whose vertices are either bags or flower vertices, and whose flower vertices are maximal anemones or daisies.

So for Conjecture 3, any attempt to keep balanced intrinsic cut rank equal to 2 across many cuts cannot appeal to arbitrary low-order behavior. It must pass through a partial-3-tree template.

Dependencies

  • [[exact-2-separation-is-2-sum.md]]
  • [[nonminimal-exact-3-separation-is-3-sum.md]]

Conflicts/Gaps

  • This theorem controls only the exact 3-separation regime, i.e. intrinsic cut rank 2.
  • It applies only to 3-connected matroids and only after discarding sequential 3-separations up to the source paper's equivalence relation.
  • Therefore it does not by itself settle the Conjecture 3 frontier, which still allows balanced cuts of larger constant or slowly growing rank.

Sources

  • 10.1016/j.jctb.2004.03.006