An equality for balanced digraphs

TL;DR

The paper proves that in balanced digraphs, the number of $k$-arc subsets that are acyclic and have a given root $s$ as a to-root is independent of the choice of $s$, generalizing spanning arborescences and acyclic orientations results. It achieves this via a general counting framework using attraction basins, cycle constraints, and a pair of inverse constructions that transfer counts between different root choices. The key contribution is a unifying invariant $\gamma_{k,Z}(s)$ (and its non-acyclic counterpart $\delta_k(s)$) across all roots and, in particular, its special cases recover classical equalities for spanning arborescences and acyclic orientations, linking to Tutte polynomial and graphic-arrangement perspectives. This provides a robust combinatorial identity with potential algorithmic and theoretical implications for balanced digraphs and related structures.

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".

Theorems & Definitions (41)

• Theorem 1.1
• Example 1.2
• Theorem 1.3
• Example 1.4
• Theorem 1.5
• Example 1.6
• proof : Proof of Theorem \ref{['thm.baldelta']} using Theorem \ref{['thm.gen']}.
• Proposition 3.1
• proof
• Lemma 3.2