Generative Structural Theory v1.29 — Generalized Theta Relations with a Direct Edge: Exact Counting and Minimality

GST v1.28 counted, for generalized theta relations -- two nodes u and v joined by r >= 3 paths with no inner node in common, each of length at least 2, one-way edges permitted -- the strongly connected spanning sub-relations (s), the inclusion-minimal ones (mu) and the least number of pairs of one (mc). This manuscript treats the case in which exactly one of the r paths is a direct edge between u and v (length 1) and the other r - 1 >= 2 paths have length at least 2; every edge of the parent relation, the direct edge included, carries one or both of its pairs. The direct edge has no inner node, so every subset of its pairs is in one of the four path states. The conditions for strong connectivity of the parent relation and of a spanning sub-relation keep the form of v1.28, with necessity and sufficiency argued separately and the proofs extended to a path of length 1 rather than applied directly; s = PROD n_i - PROD (b_i + g_i) - PROD (f_i + g_i) + PROD g_i over the r paths, the counts of the direct edge being argued for length 1. Minimality changes: a pair of the direct edge can be removed whenever a long path carries its direction. The inclusion-minimal members are classified by the state of the direct edge: empty, with the long paths forming a minimal member by themselves (type I or II of v1.28); the pair u -> v only, with every long path the exact path from v to u or a path with exactly one empty edge and both pairs on every other edge, at least one exact; the pair v -> u only, likewise with directions exchanged; both pairs, with every long path of the last kind. Minimality of each kind and necessity of the classification are proved separately, and mu follows by counting. For a strongly connected parent relation R, with L the sum of the lengths of the r paths (the direct edge counted 1) and R° the relation without the two pairs of the direct edge, mc(R) = L - 1 if R° is strongly connected and mc(R) = L otherwise, by a lower bound and a member attaining it in each case; R° fails to be strongly connected exactly when every long path carries the whole direction u -> v only, or every long path v -> u only (two-way edges on a path allowed). With v1.24 such blocks enter the decomposition of finite loop-free strongly connected components and relations, together with bridges, cycles and the generalized theta blocks of v1.28; a cycle of length at least 4 with one chord between non-adjacent nodes is such a block with r = 3. On the lengths (1, 2, 2), K_4 with one edge removed: all edges two-way, s = 228, mu = 20, mc = 4; the direct edge one-way and all other edges two-way, s = 91, mu = 11, mc = 4; both long paths one-way from u to v and the direct edge from v to u, s = 1, mu = 1, mc = 5. The members not using the direct edge are those of R°; when R° is strongly connected they agree with v1.28 for three or more long paths and with the cycle values of v1.25 for two. A related formula for the bidirected theta case is reported in a thesis by Cox (attributed there to Brown and Dilcher); its correspondence has not been checked, and no novelty is claimed. All results are argued by hand; no machine enumeration was made. Parent relations with an empty edge and general 2-connected blocks are not treated.