# Generalized edge-rankings of trees

## Abstract

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 labeling (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 optimal *c*-edge-ranking of *G*, that is, a *c*-edge-ranking using the minimum number of ranks, has applications in scheduling the manufacture of complex multi-part products; it is equivalent to finding a *c*-edge-separator tree of *G* having the minimum height. We present an algorithm to find an optimal *c*-edge-ranking of a given tree *T* for any positive integer *c* in time *O*(*n*^{2} log *δ*), where *n* is the number of vertices in *T* and *δ* is the maximum vertex-degree of *T*. Our algorithm is faster than the best algorithm known for the case *c*=1.

## Key words

Algorithm Edge-ranking Separator tree Tree## Preview

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