Minimum edge ranking spanning tree problem on block graphs

An edge ranking of a graph G is a labeling of edges of G with positive integers such that every path in G between any two edges of the same label i contains at least one edge with label j > i. The minimum edge ranking problem on G is to find an edge ranking whose largest label is smallest among all possible edge rankings of G. The minimum edge ranking spanning tree problem on G is to find a spanning tree T of G such that the minimum edge ranking of T is minimum among all possible spanning trees of G. In this paper, we propose a polynomial-time algorithm for solving the minimum edge ranking spanning tree problem on block graphs.