Abstrakt
TheWiener indexis a graph parameter originating from chemical graphtheory. It is de ned as the sum of the lengths of the shortest paths between all pairsof vertices in given graph.
In 1991,Soltes posed the following problem regardingWiener index. Find all graphs such that its Wiener index is preserved upon removalof any vertex.
The problem is far from being solved and to this day, only one suchgraph is known { the cycle graph on 11 vertices.In this paper we solve a relaxed version of the problem, proposed by Knor,Majstorovic andSkrekovski. The problem is to nd for a givenk(in nitely many)graphs such that they have exactlykvertices such that if we remove any one ofthem, the Wiener index stays the same.
We call such verticesgoodvertices and weshow that there are in nitely many cactus graphs with exactlykcycles of length atleast 7 that contain exactly 2kgood vertices and in nitely many cactus graphs withexactlykcycles of lengthc2 f5;6gthat contain exactlykgood vertices. On theother hand, we prove thatGhas no good vertex if the length of the longest cycleinGis at most 4.