Graphs preserving total distance upon vertex removal

Abstract The total distance or Wiener index W ( G ) of a connected graph G is defined as the sum of distances between all pairs of vertices in G. In 1991, Soltes posed the problem of finding all graphs G such that the equality W ( G ) = W ( G − v ) holds for all their vertices v. Up to now, the only known graph with this property is the cycle C11. Our main object of study is a relaxed version of this problem: Find graphs for which total distance does not change when a particular vertex is removed. We show that there are infinitely many graphs that satisfy this property. This gives hope that Soltes's problem may have also some solutions distinct from C11.