For a simple graph G, the 2-distance graph, D2β(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 D2β(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 G^ such that
D2β(G) is connected if and only if D2β(G^) is connected. Especially,
D2β(G) is disconnected if and only if G^ is bipartite.Comment: 7 pages, 6 figure