A central role in topological dimension theory is played by Lipscomb’s space \(J_{A}\) since it is a universal space for metric spaces of weight \(|A|\ge \aleph _{0}\) . On the one hand, Lipscomb’s space is the attractor of a possibly infinite iterated function system, i.e. it is a generalized Hutchinson-Barnsley fractal. As, on the other hand, some classical fractal sets are universal spaces, one can conclude that there exists a strong connection between topological dimension theory and fractal set theory. A generalization of Lipscomb’s space, using graphs, has been recently introduced (see Miculescu and Mihail in Aequat Math, 96:1141–1157, 2022). It is denoted by \(J_{A}^{\mathcal {G}}\) and is called the graph Lipscomb’s space associated with the graph \(\mathcal {G}\) on the set A. It turns out that it is a topological copy of a generalized Hutchinson-Barnsley fractal. This paper provides a characterization of those graphs \(\mathcal {G}\) for which \(J_{A}^{\mathcal {G}}\) is connected. In the particular case when A is finite, some supplementary characterizations are presented.
