Shahid Bahonar University of Kerman Journal of Mahani Mathematical Research 2251-7952 15 2 2026 06 01 Connectivity of 2-distance graphs 115 124 5217 10.22103/jmmr.2025.24724.1754 EN Sayyed Heidar Jafari Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran Seyed Reza Musawi Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran Journal Article 2025 01 23 For a simple graph $G$, the $2$-distance graph, $D_2(G)$, is a graph with the vertex set $V(G)$ and two vertices are adjacent if their distance is $2$ in the graph $G$. In this paper, we characterize all graphs with connected $2$-distance graphs. For graphs with diameter 2, we prove that $D_2(G)$ is connected if and only if $G$ has no spanning complete bipartite subgraphs.  For graphs whose diameter is greater than $2$, we define a maximal fine set, and by contracting $G$ with respect to these subsets, we obtain a new graph $\widehat{G}$  such that $D_2(G)$ is connected if and only if $D_2(\widehat{G})$ is connected. In particular, $D_2(G)$ is disconnected if and only if $\widehat{G}$ is bipartite. https://jmmrc.uk.ac.ir/article_5217_1777319dbbd2b056b50001e44ecbb54e.pdf