Generative Structural Theory v1.26 — Beyond a Cycle: Exact Counting and Minimality in Bidirected Theta Relations

GST v1.24 reduced three numbers of a strongly connected component -- s, the number of strongly connected spanning sub-relations; mu, the number of inclusion-minimal ones; and mc, the least number of pairs of one -- to its undirected blocks, and v1.25 computed them for cycle blocks. This manuscript takes the next block: two nodes u and v joined by three paths with no inner node in common, each of length at least 2, every edge carrying both of its pairs (a bidirected theta relation). The restriction of a strongly connected sub-relation to a path is in exactly one of four states: all forward pairs with some backward pair missing, the reverse, the full path, or exactly one empty edge with both pairs on every other edge. A sub-relation is strongly connected if and only if every path is in one of these states, some path carries the whole forward direction and some path the whole backward direction (one full path may do both). Hence s = t_1 t_2 t_3 - 2 a_1 a_2 a_3 + l_1 l_2 l_3 with t_i = 2^(l_i+1) - 1 + l_i and a_i = 2^(l_i) - 1 + l_i. The inclusion-minimal members are of two types: each path an exact directed path or a path with one empty edge, at least one in each direction; or one full path with the two other paths each having one empty edge. So mu = 6 + 2(l_1 + l_2 + l_3) + l_1 l_2 + l_1 l_3 + l_2 l_3, and mc = l_1 + l_2 + l_3, one more than the number of nodes; the members of the second type are inclusion-minimal without having the least number of pairs. With v1.24 and v1.25, components whose blocks are bridges, cycles and such theta blocks are counted. For the bidirected K2,3, s = 487, mu = 30 and mc = 6; 12 of the 18 members with 6 pairs leave one edge with no pair. A related strong-connectivity reliability formula is reported in a thesis by Cox (attributed there to Brown and Dilcher); its correspondence with the counting formula here has not been checked, and no novelty is claimed. All results are argued by hand; no machine enumeration was made. More than three paths, paths of length 1, one-way edges in the parent relation and general 2-connected blocks are not treated.