An efficient algorithm for band connectivity (BC) resoln. is presented. The method uses only readily available band coeffs. and the overlap matrix, and has a low computational cost. The accuracy of the BC resoln. is such that the method is practical for meshes of k points typically used in systems with small unit cells (e.g., 16´16´16 mesh for a 3 .ANG. unit cell). We establish that the errors in the linear tetrahedron (LT) method due to the undetected crossings have D2 dependence with respect to the characteristic spacing between k points. The intrinsic error of the LT method is proportional to D2, while for the "improved" LT method (iLT) it is proportional to D4. Thus, the BC error is in fact the leading error of the iLT method. Our benchmarks demonstrate that the resoln. of band connectivity restores the high accuracy of the "improved" LT method (D4) in systems with band crossings near or at the Fermi level.