AbstractA procedure is given for identifying the facets of the octahedron that are first intersected upon extending the edge of a polyhedral cone. The information generated by this procedure can be exploited to advantage by cut-search procedures for zero-one integer programming. Results are given which make it possible to determine the first two facets (or sets of “tied facets”) following the innermost facet with less effort than required to determine the innermost facet itself. Depending on the orientation of the extended edge relative to the octahedron, a number of successive additional facets may be determined with comparable ease.