Testing branch-width

An integer-valued function f on the set 2V of all subsets of a finite set V is a connectivity function if it satisfies the following conditions: (1) f(X) + f(Y ) ≥ f(X ∩Y )+ f(X ∪Y ) for all subsets X, Y of V , (2) f(X) = f(V \X) for all X ⊆ V , and (3) f(∅) = 0. Branch-width is defined for graphs, matroids, and more generally, connectivity functions. We show that for each constant k, there is a polynomial-time (in |V |) algorithm to decide whether the branch-width of a connectivity function f is at most k, if f is given by an oracle. This algorithm can be applied to branch-width, carving-width, and rank-width of graphs. In particular, we can recognize matroids M of branch-width at most k in polynomial (in |E(M)|) time if the matroid is given by an independence oracle.