On the complexity of all (g, f)-factors problem

Abstract Let G be a graph with vertex set V and let g , f : V → Z + be two functions such that g ≤ f . We say that G has all ( g , f ) -factors if G has an h -factor for every h : V → Z + such that g ( v ) ≤ h ( v ) ≤ f ( v ) for every v ∈ V and ∑ v ∈ V h ( v ) ≡ 0 ( mod 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, (Niessen, 1998).) 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.