Generative Structural Theory v1.88 — Beyond Bidirected Reattachment: One-Sided Leaf Moves under Budget Two

(Q19) of v1.87 asks for steps of other forms at the residual transit leaves of a tree S where the bidirected re-hanging of v1.87 fails: there a bidirected re-hanging of the leaf at its tree neighbour deletes the two pairs of the leaf and so can give up no pair of the core within the deletion bound two. This text does not settle (Q1) of v1.76, the two-deletion chain question; it treats the two re-hangings of a transit leaf at its tree neighbour that keep one pair of the leaf, fixes their budget and the exact condition of their minimality, and applies them to leaf reduction. Lemma ϗ1: let a leaf l of S, with tree neighbour p, be a transit node of a minimal strongly connected relation h, with in-neighbour a and out-neighbour b, a ≠ b, p not a or b, as at every transit leaf of a residual instance, h0 the relation h without the two pairs at l, V' the other nodes, and x either g ∪ {(p, l), (l, b)} (side 0, the pair (l, b) kept, v = (p, b)) or g ∪ {(a, l), (l, p)} (side 1, the pair (a, l) kept, v = (a, p)), g a relation on V'. Then |h minus x| = 1 + |h0 minus g|, so that within the bound two at most one pair of h0 can be given up; x is minimal exactly when (E) holds: v is not in g, g ∪ {v} is strongly connected on V', and every pair of g is essential in g ∪ {v}, the essentiality of v itself not being asked, so that (E) is not "g ∪ {v} minimal"; and l is then of the case (ii) or (iii). So the condition (O_s), that g = (h0 minus D) ∪ B with D of at most one pair of h0 and B a set of pairs of the bidirected tree PS-l of S without l satisfies (E), is necessary and sufficient for one step of side s. Theorem ϗ2 and Corollary ϗ3: under (O_s), the re-hanging is a step inside h ∪ P_S with at most two deletions to a minimal relation in which l is of the case (ii) or (iii), and with (Q1) on one node less a chain to P_S with deletion bound two follows; the procedure P_FGHJNXY^first, with a new terminal rule Y after the rule X of v1.87, gives such a chain whenever it ends at a leaf of the cases (i)-(iii), and ends at such a leaf wherever P_FGHJNX^first of v1.87 does. These statements hold for every finite node set; no structural form of (O_s) beyond its definition and no way of deciding it in polynomial time is claimed, and other forms of step and chains of several steps are not treated. Finite checks at six nodes find that, of the 482 instances of the population of v1.85 with a closed transit leaf whose neighbours are not joined by an edge of S, 364 have (C'') at none of these leaves (534 leaves); a one-sided step with at most two deletions exists at 350 of the 534 leaves, in 312 of the 364 instances, and at the other 52, where (O) and (C'') fail at every such leaf, no re-hanging at the tree neighbour of the two one-sided forms or of the bidirected form exists at these leaves within the bound two. For each side, the pair of the core from p to a (side 0) or from b to p (side 1) is in the core at 184 leaves, and 152 of these 184 have a step deleting it. P_FGHJNXY^first ends at a leaf of the cases (i)-(iii) in 20020 of the 21192 residual instances (P_FGHJNX^first in 19688; 332 gained, 312 of them by one step of the rule Y at the initial relation, none lost) and leaves 1172, 22 of class I and 1150 of class II. 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.