Average distances in undirected graphs and the removal of vertices (Q1263601)

P. M. Winkler conjectured that every connected graph G contains a vertex k such that the removal of k and all incident edges enlarges the average distance between vertices of G by at most the factor 4/3. The authors prove that every m-connected graph was a vertex whose removal increases the average distance in the graph by no more than a factor of \(m/(m-1).\) This proves Winkler's conjecture for 4-connected graphs.