Graphs preserving Wiener index upon vertex removal

The 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, Å oltés 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 Wiener index does not change when a particular vertex v is removed. In an earlier paper we have shown that there are infinitely many graphs with the vertex v of degree 2 satisfying this property. In this paper we focus on removing a higher degree vertex and we show that for any k ≥ 3 there are infinitely many graphs with a vertex v of degree k satisfying W(G)=W(G−v). In addition, we solve an analogous problem if the degree of v is n−1 or n−2. Furthermore, we prove that dense graphs cannot be a solutions of Å oltes's problem. We conclude that the relaxed version of Å oltés's problem is rich with a solutions and we hope that this can provide an insight into the original problem of Å oltés.