Edge ranking of graphs is hard

AbstractAn edge ranking of a graph is a restricted coloring of the edges with integers. It requires that every path between two edges with the same label i contains an intermediate edge with label j > i. An edge ranking is optimal if it uses the least number of distinct labels among all possible edge rankings. Recent research has revealed that the problem of finding an optimal edge ranking when restricted to trees admits a polynomial-time solution, yet the complexity of the problem for general graphs has remained open in the literature. In this paper, we prove that finding an optimal edge ranking of a graph is NP-hard. Also, we show that even finding a reasonably small edge ranking is infeasible in some cases