Transfinite Digraphs

TL;DR

This paper introduces the concept of transfinite digraphs, extending the hierarchy of digraphs to transfinite ranks using ordinal-based constructions, including special 'arrow rank' digraphs for limit ordinals.

Contribution

It is the first to define and analyze transfinite digraphs, establishing a hierarchy of ranks indexed by countable ordinals and introducing 'arrow rank' digraphs for limit stages.

Findings

Established a hierarchy of transfinite digraphs by ordinal rank.

Defined 'arrow rank' digraphs for limit ordinal stages.

Extended the concept of digraphs beyond finite and natural-number ranks.

Abstract

Transfinite graphs have been defined and examined in a variety of prior works, but transfinite digraphs had not as yet been investigated. The present work embarks upon such a task. As with the ordinals, transfinite digraphs appear in a hierarchy of ranks indexed by the countable ordinals. The digraphs of rank 0 are the conventional digraphs. Those of rank 1 are constructed by defining certain extremities of 0-ranked digraphs, and then partitioning those extremities to obtain vertices of rank 1. Then, digraphs of rank 0 are connected together at those vertices of rank 1 to obtain a digraph of rank 1. This process can be continued through the natural-number ranks. However, to achieve a digraph whose rank is the first infinite ordinal $\omega$ (i.e., the first limit ordinal), a special kind of transfinite digraph, which we call a digraph with an "arrow rank" must first be constructed in a way different from those of natural-number rank. Then, digraphs of still higher ranks can be constructed in a way similar to that for the natural-number ranked digraphs. However, just before each limit-ordinal rank, a digraph of arrow rank must be set up.