Solution to an open problem on the closeness of graphs

Abstract

A network can be analyzed by means of many graph theoretical parameters. In the context of networks analysis, closeness is a structural metric that evaluates a node's significance inside a network. A cactus is a connected graph in which any block is either a cut edge or a cycle. This paper analyzes the closeness of cacti, we determine the unique graph that minimizes the closeness over all cacti with fixed numbers of vertices and cycles, which solves an open problem proposed by Poklukar \& \v{Z}erovnik [Fundam. Inform. 167 (2019) 219--234]