A relaxed version of \v{S}olt\'{e}s's problem and cactus graphs

TL;DR

This paper investigates a relaxed version of Soltes's problem on the Wiener index, demonstrating the existence of infinitely many cactus graphs with specific properties related to vertex removal and cycle lengths.

Contribution

It introduces a new class of graphs with a fixed number of vertices maintaining the Wiener index upon vertex removal, expanding understanding of graph structures related to Soltes's problem.

Findings

Infinitely many cactus graphs with k cycles of length ≥7 have exactly 2k good vertices.

Infinitely many cactus graphs with k cycles of length 5 or 6 have exactly k good vertices.

No good vertices exist in graphs with maximum cycle length at most 4.

Abstract

The \emph{Wiener index} is one of the most widely studied parameters in chemical graph theory. It is defined as the sum of the lengths of the shortest paths between all unordered pairs of vertices in a given graph. In 1991, \v{S}olt\'es posed the following problem regarding the Wiener index: Find all graphs such that its Wiener index is preserved upon removal of any vertex. The problem is far from being solved and to this day, only one graph with such property is known: the cycle graph on 11 vertices. In this paper, we solve a relaxed version of the problem, proposed by Knor et al.\ in 2018. For a given $k$, the problem is to find (infinitely many) graphs having exactly $k$ vertices such that the Wiener index remains the same after removing any of them. We call these vertices \emph{good} vertices and we show that there are infinitely many cactus graphs with exactly $k$ cycles of length at least 7 that contain exactly $2k$ good vertices and infinitely many cactus graphs with exactly $k$ cycles of length $c \in \{5,6\}$ that contain exactly $k$ good vertices. On the other hand, we prove that $G$ has no good vertex if the length of the longest cycle in $G$ is at most $4$.