Generative Structural Theory v1.34 — Loopless Multigraph Subdivisions with One-Way Edges: Projection Formulas for Counting and Minimality

The parent relation is a subdivision of a finite connected loopless multigraph H with at least two vertices and distinguished edges (each edge replaced by a path of length l_e >= 1, inner nodes not shared, at most one edge of length 1 between any two vertices), in which every edge of every path carries one or both of its pairs; for its spanning sub-relations this manuscript counts the strongly connected ones (s), the inclusion-minimal ones (mu) and the least number of pairs of one (mc). Under these conditions, every parent relation already admits a description within v1.33; the extension permits a coarser base graph with fewer base vertices and distinguished parallel edges. Parallel edges are kept apart by a labelled projection: each edge of H gives two labelled arcs with opposite readings, and the projection of a sub-relation holds the arc of an edge in a direction when its path carries every pair of that direction. Under the condition that the sub-relation on every path is in one of the four path states of v1.33, strong connectivity of the whole is decided by the trace of the projection on the base vertices; the parent is strongly connected iff no path has one-way edges of both directions and its projection is strongly connected, otherwise s = mu = 0 and mc is not defined. s is a weighted sum over the strongly connected labelled projections, and a cut inclusion-exclusion with one factor per edge, parallel edges giving separate factors. The minimality criterion of v1.33 holds for strongly connected sub-relations with essentiality read on labelled arcs: an arc is not essential when a parallel arc with the same reading remains, and an example with two parallel paths shows that the criterion read on the merged relation gives a wrong verdict. mu is a sum over the admissible labelled projections and mc the least cost over the feasible ones. For two base vertices s and mu take closed forms (types I and II of the minimal members), and for a strongly connected parent with at least two paths mc is the total length L, or L - 1 when a path of length 1 exists and two other paths, distinct, admit the pure forward and the pure backward path respectively. The blocks of H are the classes of the equivalence generated by lying on a common cycle, cycles of two parallel edges included; the blocks of the subdivision are the subdivisions of the classes with at least two edges and the single edges of the bridge paths, so the block decomposition of v1.24 applies. With two base vertices the general formulas reduce, as formulas and not only as values, to those of the cycles of v1.25 and the theta relations of v1.26-v1.29; for simple H everything reduces to v1.33. Examples, counted by hand, with every edge two-way unless said otherwise: two parallel paths of lengths 1 and 5 (s = 133, mu = 8, mc = 6) and with the edge of length 1 one-way (s = 33, mu = 2, mc = 6); three parallel paths of length 2 (s = 487, mu = 30, mc = 6); paths of lengths 1, 2 and 2 (s = 228, mu = 20, mc = 4); the example of three parallel paths of length 2 with a two-way pendant path of length l added (s = 487, mu = 30, mc = 6 + 2l); and the same parent as the example of lengths 1, 2 and 2, described with three base vertices and recounted in that description (s = 228, mu = 20, mc = 4). No novelty is claimed. No machine enumeration or machine numerical computation was made; the proofs and counts are given by hand.