Testing branch-width

An integer-valued function f on the set 2 V 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.