The NP-completeness of Edge Ranking

Abstract

An edge ranking of a graph G is a labeling of its edges with positive integers such that very path between two edges with the same label I contains an intermediate edgy with label j > i. An edge ranking is optimal if it uses the least number of distinct labels among all possible edge rankings. Such a ranking corresponds to a minimum-height edge-separator tree of G. The problem of finding an optimal edge ranking has been studied intensively；recent development has shown that the problem when restricted to trees in not NP-hard and indeed admits a polynomial-time solution, yet the complexity of the problem for general graphs has remained open in the literature. In this paper we settle this open question and prove that finding an optimeal edge ranking of a graph is NP-hard.