Generative Structural Theory v1.86 — Beyond Condition (C): Cut Conditions for Tree-Path Leaf Reattachment

(Q17) of v1.85 asks, in its first half, for structural conditions under which a closed transit leaf of a tree S can be re-hung along the path of S between its two neighbours when the condition (C) of v1.85 fails, a part of the pairs of the path being put in so that the relation left keeps all the other pairs of h. This text does not settle (Q1) of v1.76, the two-deletion chain question; it gives a characterisation of the condition (C') of v1.85 by cut sets, and its application to leaf reduction. Lemma Ͼ1: let a leaf l of S be a transit node of a minimal strongly connected relation h, with in-neighbour a and out-neighbour b, a ≠ b, A the pairs of the path of S from a to b, V' the other nodes and h0 the relation h without the two pairs at l. A cut set for a pair e of h0 is a nonempty proper subset of V' that e is the only pair of h0 to leave; for a family of cut sets, one for each pair of h0, B_U is the set of the pairs of A not in h0 that leave none of them. The condition (D), some family of cut sets with h0 ∪ B_U strongly connected on V', holds exactly when (C') holds, some set B of pairs of A not in h0 with h0 ∪ B minimal on V'; a family gives such a B by taking out pairs of B_U in the order of the path, and a set B gives a family by the nodes reached from the tail of each pair e of h0 in (h0 ∪ B) minus {e}; and (C) implies (D). Theorem Ͼ2 and Corollary Ͼ3: at a closed transit leaf with (D), hanging l again at its tree neighbour p by the 2-cycle of l and p gives, by v1.85 Lemma ϒ1 (c), a minimal relation inside h ∪ P_S with at most two deletions in which l is of the cases (i), (ii) and (iii), and with (Q1) on one node less a chain to P_S with deletion bound two; the procedure P_FGHJN^first, with a new terminal rule N at the leaves where (C) fails and (C') holds, gives such a chain whenever it ends at a leaf of the cases (i)-(iii), and ends at such a leaf wherever P_FGHJ^first of v1.85 does. These statements hold for every finite node set; no way of deciding (D) in polynomial time is claimed. Finite checks at six nodes find that, of the 1484 instances of the population of v1.85, 482 have a closed transit leaf whose neighbours are not joined by an edge of S, (C) holds at one of them in 98 and (C') in 110, and the 12 with (C') at one of them and (C) at none have one such leaf and one set B each. P_FGHJN^first ends at a leaf of the cases (i)-(iii) in 19680 of the 21192 residual instances (P_FGHJ^first in 19668; the 12 gained all by one step of the rule N at the initial relation, none lost) and leaves 1512, 26 of class I and 1486 of class II. At every transit leaf with a ≠ b of every minimal relation with three to six nodes and one tree of each shape, (D), decided from its definition, and (C') agree. A success counted here is an arrival at a leaf of the cases (i)-(iii), not an instance for which (Q1) is proved. These numbers are findings of the six-node enumeration, not general statements.