Acyclic orientations with degree constraints

TL;DR

This paper investigates the complexity of generalized acyclic orientations with degree constraints, showing polynomial solvability under certain conditions and NP-completeness in others, thus answering a longstanding open question.

Contribution

It characterizes the computational complexity of degree-constrained acyclic orientations, providing a complete dichotomy for specific function conditions.

Findings

Polynomial-time solvable when $fg ot eq 0$

NP-complete when $f eq 0$ and $g eq 0$

Answers a question posed in 2009

Abstract

In this note we study the complexity of some generalizations of the notion of $st$-numbering. Suppose that given some functions $f$ and $g$, we want to order the vertices of a graph such that every vertex $v$ is preceded by at least $f(v)$ of its neighbors and succeeded by at least $g(v)$ of its neighbors. We prove that this problem is solvable in polynomial time if $fg\equiv 0$, but it becomes NP-complete for $f\equiv g \equiv 2$. This answers a question of the first author posed in 2009.