Graphs whose Wiener index does not change when a specific vertex is removed

The Wiener index W(G) of a connected graph G is defined to be the sum of distances between all pairs of vertices in G. In 1991, Soltes studied changes of the Wiener index caused by removing a single vertex. He posed the problem of finding all graphs G so that equality W(G) = W(G - v) holds for all their vertices v. The cycle with 11 vertices is still the only known graph with this property. In this paper we study a relaxed version of this problem and find graphs which Wiener index does not change when a particular vertex v is removed. We show that there is a unicyclic graph G on n vertices with W(G) = W(G - v) if and only if n >= 9. Also, there is a unicyclic graph G with a cycle of length c for which W(G) = W(G - v) if and only if c >= 5. Moreover, we show that every graph G is an induced subgraph of H such that W(H) = W(H - v). As our relaxed version is rich with solutions, it gives hope that Soltes's problem may have also some solutions distinct from C-11. (C) 2017 Elsevier B.V. All rights reserved.