An equality for balanced digraphs

TL;DR

This paper proves a new combinatorial equality for balanced directed multigraphs, relating counts of certain acyclic subgraphs with specified path properties, generalizing classical results in graph theory.

Contribution

It establishes a novel equality for counts of specific acyclic subgraphs in balanced digraphs, extending known theorems about arborescences and feedback arc sets.

Findings

Number of $s$-convergences is independent of vertex $s$

Generalizes classical results on spanning arborescences and acyclic orientations

Extends to subgraphs with prescribed cycle sets

Abstract

Consider a directed multigraph $D$ that is balanced (i.e., at each vertex, the indegree equals the outdegree). Let $A$ be its set of arcs. Fix an integer $k$. Let $s$ be a vertex of $D$. We show that the number of $k$-element subsets $B$ of $A$ that contain no cycles but contain a path from each vertex to $s$ (we call them "$s$-convergences") is independent on $s$. This generalizes known facts about spanning arborescences, acyclic orientations and maximal acyclic subdigraphs (or, equivalently, minimum feedback arc sets). Moreover, this result can be generalized even further, replacing "contain no cycles" with "have a given set of cycles".