Greedily constructing maximal partial f-factors

Let G=(V,E) be a graph and let f be a function f:V→N. A partial f-factor of G is a subgraph H of G, such that the degree in H of every vertex v∈V is at most f(v). We study here the recognition problem of graphs, where all maximal partial f-factors have the same number of edges. Graphs which satisfy that property for the function f(v)≡1 are known as equimatchable and their recognition problem is the subject of several previous articles in the literature. We show the problem is polynomially solvable if the function f is bounded by a constant, and provide a structural characterization for graphs with girth at least 5 in which all maximal partial 2-factors are of the same size.