When all directed cycles have the same weight

TL;DR

This paper characterizes the structure of weightable directed graphs, showing how they can be constructed from simpler classes and providing a polynomial-time algorithm to test weightability.

Contribution

It introduces constructions that generate all weightable digraphs from planar and circular digraphs, and presents an efficient testing algorithm.

Findings

Every planar weightable digraph can be constructed from circular digraphs.

Every weightable digraph can be built from planar digraphs.

A polynomial-time algorithm exists to determine if a digraph is weightable.

Abstract

A digraph $G$ is weightable if its edges can be weighted with real numbers such that the total weight in each directed cycle equals 1. There are several equivalent conditions: that $G$ admits a 0/1-weighting with the same property, or that $G$ contains no subdivided "double-cycle" as a subdigraph, or that for every triple of vertices, all directed cycles containing all three pass through them in the same cyclic order. And there is quite a rich supply of such digraphs: for instance, any digraph drawn in the plane such that each of its directed cycles rotates clockwise around the origin is weightable (let us call such digraphs "circular"), and there are weightable planar digraphs with much more complicated structure than this. Until now the general structure of weightable digraphs was not known, and that is our objective in this paper. We will show that: - there is a construction that builds every planar weightable digraph from circular digraphs; and - there is a (different) construction that builds every weightable digraph from planar ones. We derive a poly-time algorithm to test if a digraph is weightable.