Greedily constructing maximal partial -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.