Some inequalities for orderings of acyclic digraphs

TL;DR

This paper establishes an inequality relating the sum of differences between indegree and outdegree of vertices in acyclic digraphs to their orderings, characterizing when equality occurs for certain poset comparability graphs.

Contribution

It proves a new inequality for acyclic digraphs involving their acyclic orderings and characterizes the cases of equality as comparability digraphs of posets with order dimension two.

Findings

The inequality holds for all acyclic orderings.

Equality characterizes comparability digraphs of posets with order dimension two.

Provides a new perspective on the structure of acyclic digraphs.

Abstract

Let $D=(V,A)$ be an acyclic digraph. For $x\in V$ define $e_{_{D}}(x)$ to be the difference of the indegree and the outdegree of $x$. An acyclic ordering of the vertices of $D$ is a one-to-one map $g: V \rightarrow [1,|V|] $ that has the property that for all $x,y\in V$ if $(x,y)\in A$, then $g(x) < g(y)$. We prove that for every acyclic ordering $g$ of $D$ the following inequality holds: \[\sum_{x\in V} e_{_{D}}(x)\cdot g(x) ~\geq~ \frac{1}{2} \sum_{x\in V}[e_{_{D}}(x)]^2~.\] The class of acyclic digraphs for which equality holds is determined as the class of comparbility digraphs of posets of order dimension two.