# The Complexity of All $(g,f)$-Factor Problem

## TL;DR

This paper proves that determining whether a graph has all $(g,f)$-factors is NP-hard, answering a longstanding open question and showing the problem's computational complexity.

## Contribution

It establishes the NP-hardness of testing for all $(g,f)$-factors in graphs, providing a negative answer to Niessen's 1998 open problem.

## Key findings

- Proves the problem is NP-hard.
- Answers Niessen's open question negatively.
- Highlights computational difficulty of all $(g,f)$-factor problem.

## Abstract

Let $G$ be a graph with vertex set $V$ and let $g, f : V\rightarrow \mathbb{Z}^+$ be two functions such that $g\le f$. We say that $G$ has all $(g, f )$-factors if $G$ has an $h$-factor for every $h: V\rightarrow \mathbb{Z}^+$ such that $g(v)\le h(v)\le f (v)$ for every $v\in V$ and $\sum_{v\in{V}}h(v)\equiv 0\pmod 2$. Two decades ago, Niessen derived from Tutte's $f$-factor theorem a similar characterization for the property of graphs having all $(g, f )$-factors and asked whether there is a polynomial time algorithm for testing whether a graph $G$ has all $(g, f )$-factors (A characterization of graphs having all $(g, f )$-Factors, \emph{J. Combin. Theory, Ser. B}, \textbf{72} (1998), 152--156). In this paper, we show that it is NP-hard to determine whether a graph $G$ has all $(g,f)$-factors, which gives a negative answer to the question of Niessen.