Generative Structural Theory v1.25 — Inside a Block: Exact Counting for Cycle-Supported Relations

GST v1.24 reduced the 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 the same numbers for its undirected blocks, a bridge giving 1, 1 and 2. This manuscript computes them inside the next simplest block: a strongly connected loop-free relation R whose undirected graph is a simple cycle on n >= 3 nodes, each edge carrying one or both of its two pairs. Strong connectivity of R forces all one-way edges to point the same way round the cycle; when there are one-way edges, that common way is called forward (when every edge is two-way, either way may be called forward). A sub-relation W of R is strongly connected if and only if it contains the forward directed cycle, or the backward one, or consists of both pairs on every edge but one, with that edge empty; the first two cases overlap only in R itself when every edge is two-way. From this classification the inclusion-minimal members are listed completely: the forward cycle, the backward cycle when every edge is two-way, and the paths with both pairs on all edges but one whenever those lie in R. With k the number of one-way edges of R: for k = 0, s = 2^(n+1) + n - 1, mu = n + 2 and mc = n; for k = 1, s = 2^(n-1) + 1, mu = 2 and mc = n; for 2 <= k <= n, s = 2^(n-k), mu = 1 and mc = n. The two-way paths are inclusion-minimal without having the least number of pairs. The count for k = 0 follows from a known formula for the strong-connectivity reliability of the bidirected cycle, and no novelty is claimed for it; the proof here is self-contained. With v1.24, every strongly connected component whose blocks are bridges and cycles is counted: s and mu are products over the cycle blocks and mc = 2b + SUM n_i for b bridges and cycles of lengths n_i. The triangle block of the fixed cut z_t (n = 3, k = 1) gives the published values 5, 2 and 3, and the bidirected triangle of v1.24 gives 18, 5 and 3; these are applications to published examples. All results are argued by hand; no machine enumeration was made. General 2-connected blocks are not treated.