Characterizing 2-distance graphs

Let X be a finite simple graph. The 2-distance graph D2(X) of X is the graph with the same vertex set as X and two vertices are adjacent if and only if their distance in X is exactly 2. A graph G is a 2-distance graph if there exists a graph X such that D2(X)≅G. In this paper, we give three characterizations of 2-distance graphs, and find all graphs X such that D2(X)≅kP2 or Km∪Kn, where k≥2 is an integer, P2 is the path of order 2, and Km is the complete graph of order m≥1