When all directed cycles have length three

TL;DR

The paper tackles the problem of characterizing all $3$-cyclic digraphs, building a constructive framework that decomposes any such digraph into simpler units. It introduces and exploits concepts like $3$-rings, $3$-annular drawings, diwheels, and the special-edge-sum operation to assemble unbreakable $3$-cyclic digraphs from $3$-pinched and diwheel-free components, with a detailed ear-decomposition theory in the diwheelfree case. It also provides equivalent characterizations and obstructions for $3$-annularity and develops a complete weightability theory: a digraph is weightable if and only if no subdigraph is a weak $k$-double-cycle for $k\ge3$, and such weightings can be taken in $\{0,1\}$-valued form. Together, these results yield a modular, constructive blueprint for recognizing and building $3$-cyclic digraphs and for understanding the cycle-structure constraints via weightability.

Abstract

We give a construction to build all digraphs with the property that every directed cycle has length three.

Figures (3)

• Figure 1: An $\ell$-ring.
• Figure 2: A $4$-annular drawing.
• Figure 3: Two of the branchers