When all directed cycles have the same weight

TL;DR

This work provides a complete structural theory for weightable digraphs, where every directed cycle has total weight 1, by proving two complementary constructions: planar weightable digraphs can be built from circular embeddings, and general weightable digraphs can be assembled from planar ones. It combines planar cut-style decompositions, bond carvings with diwidth two, and nonplanar composition rules to reduce any 1-strong weightable digraph to simpler, well-understood components, backed by a polynomial-time weightability test. The approach leverages ear-basis techniques, Pfaffian-brace theory, and intricate path lemmas to trace how cycles, paths, and cuts interact under these constructions. The results yield both a deep structural classification and practical algorithms for recognizing and constructing weightable digraphs, with implications for related cycle-weighting problems and graph drawing representations.

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.

Figures (23)

• Figure 1: A weightable digraph with no circular drawing. The four thick edges have weight one, and the others have weight zero.
• Figure 2: A diplanar, 2-strong, weightable digraph, that cannot be drawn in the plane such that all its dicycles are clockwise. The thick edges have weight one.
• Figure 3: A nonplanar, 2-strong, weightable digraph. Again, the thick edges have weight one.
• Figure 4: An example of a weak 4-double cycle.
• Figure 5: Construction for building non-2-strong weightable digraphs.

Theorems & Definitions (28)

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