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Well-Positioned Nonsequential 3-Separation Has Safe Element

Claim/Theorem

Let \((A,B)\) be a nonsequential 3-separation in a 3-connected matroid \(M\).

Assume:

  1. \(B\) is fully closed;
  2. \(A\) meets no triangle or triad of \(M\);
  3. if \((X,Y)\) is any nonsequential 3-separation of \(M\), then
\[ A\subseteq \operatorname{fcl}(X) \qquad\text{or}\qquad A\subseteq \operatorname{fcl}(Y). \]

Then \(A\) contains an element \(e\) such that deletion of \(e\) from \(M\) or from \(M^\ast\) is 3-connected and does not expose any new 3-separations.

Equivalently, a leaf-like, fully closed nonsequential 3-separator satisfying the source paper's technical hypotheses is inductively peelable without creating new 3-separation types.

Dependencies

  • None.

Conflicts/Gaps

  • The theorem is technical and local. It does not by itself prove that every 3-connected matroid has such a safe element.
  • Its hypotheses are tailored to a well-positioned nonsequential 3-separation, so an additional theorem is still needed to guarantee access to this regime from an arbitrary low-rank obstruction.
  • For Conjecture 3, it only addresses the intrinsic cut-rank-2 nonsequential branch.

Sources

  • 10.1016/j.aam.2010.10.009