GENERALIZED EDGE-RANKINGS OF TREES

In this paper we newly define a generalized edge-ranking of a graph G as follows: for a positive integer c, a c-edge ranking of G is a label ing (ranking) of the edges of G with integers such that, for any label i, deletion of all edges with labels > i leaves connected components, each having at most c edges with label i. The problem of finding an o ptimal c-edge-ranking of G, that is, a c-edge-ranking using the minimu m number of ranks. has applications in scheduling the manufacture of c omplex multi-part products; it is equivalent to finding a c-edge-separ ator tree of G having the minimum height. We present an algorithm to f ind an optimal c-edge-ranking of a given tree T for any positive integ er c in time O(n(2) log Delta), where n is the number of vertices in T and Delta is the maximum vertex-degree of T. Our algorithm is faster than the best algorithm known For the case c = 1.