Aparallel digraphs and splicing machines

The concepts of a splicing machine and of an aparalled digraph are introduced. A splicing machine is basically a means to uniquely obtain all circular sequences on a finite alphabet by splicing together circular sequences from a small finite set of “generators”. The existence and uniqueness of the central object related to an aparallel digraph, the strong component, is proved, and this strong component is shown to be the unique fixed point of a natural operator defined on sets of vertices of the digraph. A digraph is shown to be a splicing machine if and only if it is the strong component of an aparallel digraph. Motivation comes, on the applied side, from the splicing of circular sequences on a finite alphabet and, on the theoretical side, from the Banach fixed point theorem.