Efficient algorithms for finding the most vital edge of a minimum spanning tree

Let G( V, E) be an undirected graph with m edges and n vertices such that each edge e has a real valued weight w(e). Let MST(G) be a minimum spanning tree in G. Let ƒ( G) be the weight of a minimum spanning tree of G if G is connected; otherwise ƒ(G)∞. We define a most vital edge with respect to a minimum spanning tree in a connected undirected graph G as an edge e such that ƒ(G−e)⩾ƒ(G−e′) for every edge e′ in G. In this paper, we give O( m+n log n) and O( mα(m,n)) time algorithms, which improve O( m log m) and O( n 2) time bounds by Hsu et al. in Inform. Process. Lett. 39 (1991) 277–281.