Subgraphs versus Orientations: Infinite families of equidistributions

TL;DR

This paper generalizes classical graph enumeration results by establishing broad identities between subgraphs and orientations under various connectivity constraints, including equivalence classes.

Contribution

It introduces a unified framework connecting subgraphs and orientations with complex connectivity and equivalence constraints, extending classical enumerative results.

Findings

Established equinumerous sets of orientations and subgraphs under connectivity constraints.

Extended results to equivalence classes of orientations under cycle and cocycle reversals.

Generalized classical enumeration results to broader connectivity and equivalence conditions.

Abstract

A classical enumerative result states that, given a graph $G$ and a vertex $u$, the number of connected subgraphs of $G$ is equal to the number of orientations of $G$ such that every vertex can reach $u$ by a directed path. We show that this result is an instance of a much broader set of enumerative identities between subgraphs and orientations corresponding to various connectivity constraints. Namely, given two sets of pairs of vertices $A=\{(u_i,v_i), i\in[k]\}$ and $B=\{(u_i',v_i'), i\in[l]\}$, we consider the orientations $\alpha$ of $G$ such that adding the elements of $A$ and $B$ as additional directed edges to $\alpha$ gives an orientation $\alpha'$ in which $v_i$ cannot reach $u_i$ for all $i\in[k]$, but $v_i'$ can reach $u_i'$ for all $i\in[l]$. We show that this set of orientations is equinumerous to a set of subgraphs satisfying the ``same" connectivity constraints defined in terms of $A$ and $B$. We also extend our results to the enumeration of equivalence classes of orientations satisfying such connectivity constraints. Precisely, we consider the equivalence classes under cycle reversal, cocycle reversal, or cycle-cocycle reversal. We show that the equivalences classes are equinumerous to some sets of subgraphs defined by connectivity and acyclicity constraints.