Generative Structural Theory v1.79 — Structural Conditions for Preparatory Chains in Leaf Detachment

v1.78 gave, assuming the two-deletion chain question (Q1) of v1.76 on one node less, a chain of minimal strongly connected relations inside h ∪ P_S from h to the bidirected tree P_S, with at most two pairs deleted at each step, for every instance with a leaf of one of four cases (i)-(iv), and whenever such a chain leads from h to a relation with such a leaf; it left the instances with no such leaf ((Q5) of v1.78). This text does not settle (Q1); it gives partial results. Under their stated hypotheses, two local operations each delete at most two pairs and preserve inclusion-minimal strong connectivity within h ∪ P_S. Lemma A_Ϩ re-hangs a transit node at a neighbour in S, provided its in-neighbour reaches its out-neighbour without passing through that node. Lemma B_Ϩ moves a satellite a of a node l using a tree edge {a,b}, where b is distinct from a and l and (l,b) is in h; it also has a reversed form. Theorem H_Ϩ gives a conditional construction on any finite set V of m nodes. Let h be an inclusion-minimal strongly connected relation on V, let S be a tree on V, and let l be a leaf of S with neighbour p. Suppose (p,l) is in h and T = {w ≠ p : (w,l) is in h} is non-empty. If every w in T has exactly one incoming pair (u_w,w), with u_w outside T, and exactly one outgoing pair (w,l), and h[V minus T] is strongly connected, the nodes of T can be re-hung in |T| ≤ m−2 steps, each deleting at most two pairs, so that (p,l) is the only pair entering l. If (Q1) holds for every instance on m−1 nodes, this preparation extends to a chain from h to P_S inside h ∪ P_S with deletion bound two. Corollary Ϩ1 states the same for any steps of the two lemmas that lead to a relation with a leaf of the cases (i)-(iv), the existence of such steps, and in case (iv) of the auxiliary relation k of v1.78, being hypotheses. A finite check at six nodes finds, of the 2678 cases left by v1.78, 410 satisfying the conditions of Theorem H_Ϩ and 2634 reached by one or two steps of the lemmas, the two counts overlapping, with the whole chains checked step by step; 44 cases are explained by neither, each having a one-step preparation found by a search. It is not claimed that every residual case has a preparation of at most two steps.