Paths in graphs, reducing the edge-connectivity only by two (Q1066914)

For distinct vertices a and b in a finite, connected, undirected graph G, let \(\lambda\) (a,b;G) denote the maximum number of edge disjoint paths joining a and b. The following major result is presented. For any fixed pair of distinct vertices \(x_ 1\) and \(y_ 1\) in G there exists a path P joining \(x_ 1\) and \(y_ 1\) such that for all pairs of vertices x and y in G we have \(\lambda\) (x,y;G-E(P))\(\geq \lambda (x,y;G)-2\) where E(P) denotes the edge set of P.