Generative Structural Theory v1.28 — Generalized Theta Relations with One-Way Edges: Exact Counting and Minimality

GST v1.26 and v1.27 counted, for theta relations -- two nodes u and v joined by three paths with no inner node in common, each of length at least 2 -- the strongly connected spanning sub-relations (s), the inclusion-minimal ones (mu) and the least number of pairs of one (mc), with every edge of the parent relation carrying both of its pairs (v1.26) or one or both (v1.27). This manuscript allows any number r >= 3 of such paths, one-way edges permitted. The parent relation is strongly connected if and only if on each path the one-way edges all point the same way, some path carries the whole direction from u to v and some path the whole direction from v to u. A spanning sub-relation is strongly connected if and only if its restriction to every path is in one of the four path states of v1.26, some path carries the whole direction from u to v and some the whole direction from v to u; necessity and sufficiency are argued separately. The per-path counts (f_i, b_i, d_i, g_i) of v1.27 carry over, and s = PROD n_i - PROD (b_i + g_i) - PROD (f_i + g_i) + PROD g_i over the r paths, with n_i = f_i + b_i + d_i + g_i. For every r the inclusion-minimal members are of two types: each path an exact forward path, an exact backward path or a path with exactly one empty edge and both pairs on every other edge, with at least one path in each direction (so at most r - 2 paths in this last state); or one fully bidirected path with every other path having exactly one empty edge and both pairs on every other edge. Each member of the two types is shown minimal by single-pair removal, and every minimal member is shown to be of one of them; mu follows by counting. mc = l_1 + ... + l_r = |V| + r - 2 for every strongly connected parent relation of the class, by a lower bound from each path and a member attaining it. With v1.24 and v1.25, finite loop-free strongly connected components whose undirected blocks are bridges, cycles and such generalized theta blocks are counted, and so are finite loop-free relations each of whose strongly connected components is of this kind. For the bidirected K_{2,4}, s = 5327, mu = 142 and mc = 8; with one one-way edge, s = 1695, mu = 67 and mc = 8. For r = 3 the formulas are those of v1.27; read at r = 2 they agree with the cycle values of v1.25 for cycles of length at least 4. A related formula for the bidirected 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, paths of length 1 and general 2-connected blocks are not treated.