Testing branch-width

An integer-valued function f on the set V2 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.