On the Complexity of Counting Orderings in Graphs

2026-06-27

AI Summary

This study addresses the computational complexity of counting vertex or edge orderings in graphs subject to adjacency constraints. The authors propose a unified approach that constructs a parameterized family of graphs $G_q$ such that the counting function can be expressed as a rational function in $q$. By analyzing symbolic limits of this function, they recover known #P-hard problems and thereby establish the #P-completeness of several ordering-counting problems without relying on classical reduction techniques. This method yields the first proofs—or alternative proofs—of #P-completeness for five classes of problems, including vertex orderings in bipartite graphs, st-numberings, shellings, and linear extensions of two types of posets. Notably, it resolves a conjecture by Felsner and Manneville concerning the #P-completeness of counting linear extensions of N-free posets.

Abstract

We study the computational complexity of several counting problems on graphs. Each of these problems consists of counting orderings of the vertices or edges with adjacency constraints. We show $\#P$-completeness for all of them via a common new technique. Given a counting function $C$ of interest, we define a parameterized family of instances $G_q$, where the parameter $q$ controls the amplification of a simple gadget. After multiplying by an explicit factor $f(q)$, we show that the values of $f(q) \cdot C(G_q)$, for positive integers $q$, agree with a rational function in $q$ whose numerator and denominator can be interpolated in polynomial time. We then recover a $\#P$-hard function by evaluating this rational function symbolically at a limiting value $L \in \mathbb{Q} \cup \{\infty, -\infty\}$. With this methodology, we show $\#P$-completeness for the following counting problems: (a) successive vertex orderings of bipartite graphs, (b) st-numberings of graphs, (c) shellings of bipartite graphs, (d) linear extensions of N-free posets of height $3$, and (e) linear extensions of posets of height $2$. Result (d) settles a conjecture of Felsner and Manneville (2015). Although result (e) was first proved by Dittmer and Pak (2018), we include an alternative proof, using our technique, that does not rely on the result of Brightwell and Winkler (1991) about the hardness of counting linear extensions for general posets.

Problem

• counting complexity
• graph orderings
• #P-completeness
• adjacency constraints
• linear extensions

Innovation

• #P-completeness
• counting orderings
• rational interpolation
• parameterized gadget
• linear extensions