We describe new algorithms for determining the adjacencies between zero-dimensional cells and those one-dimensional cells that are sections (not sectors) in cylindrical algebraic decompositions (cad). Such adjacencies constitute a basis for determining all other cell adjacencies. Our new algorithms are local, being applicable to a specified OD cell and the 1D cells described by specified polynomials. Particularly efficient algorithms are given for the OD cells in spaces of dimensions two, three and four. Then an algorithm is given for a space of arbitrary dimension. This algorithm may on occasion report failure, but it can then be repeated with a modified isolating interval and a likelihood of success.