Pappus-Desargues digraph confrontation

Like the Coxeter graph became reattached into the Klein graph in [3], the Levi graphs of the 93 and 103 self-dual configurations, known as the Pappus and Desargues (k-transitive) graphs P and D (where k = 3), also admit reattachments of the distance-(k − 1) graphs of half of their oriented shortest cycles via orientation assignments on their common (k−1)-arcs, concurrent for P and opposite for D, now into 2 disjoint copies of their corresponding Menger graphs. Here, P is the unique cubic distance-transitive (or CDT) graph with the concurrent-reattachment behavior while D is one of 7 CDT graphs with the opposite-reattachment behavior, that include the Coxeter graph. Thus, P and D confront each other in these respects, obtained via C-ultrahomogeneous graph techniques [4, 5] that allow to characterize the obtained reattachment Menger graphs in the same terms. 1 Preliminaries Given a collection C of (di)graphs closed under isomorphisms, a (di)graph G is said to be C-ultrahomogeneous (or C-UH) [4, 5] if every isomorphism between 2 induced members of C in G extends to an automorphism of G. If C = {H} is the isomorphism class of a (di)graph H , we say that such a G is {H}-UH (or H-UH). In [5], C-UH graphs are studied when C is the collection of either (a) the complete graphs, or (b) the disjoint unions of complete graphs, or (c) the complements of those unions. We consider any undirected graph G as a digraph by taking each edge e of G as a pair of oppositely oriented (or O-O) arcs ~e and (~e). Then cohering) (or fastening, or zipping) ~e and (~e) (meaning that we take the