  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 and only if their distance
is $2$ in the graph $G$. In this paper, we characterize all graphs with
connected 2-distance graph. 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 with a diameter greater than 2, we define a maximal Fine set and by
contracting $G$ on these subsets, we get a new graph $\hat G$ such that
$D_2(G)$ is connected if and only if $D_2(\hat G)$ is connected. Especially,
$D_2(G)$ is disconnected if and only if $\hat G$ is bipartite.
