Characterizing $2$-Distance Graphs and Solving the Equations $T_2(X)=kP_2$ or $K_m \cup K_n$

Let $X$ be a finite, simple graph with vertex set $V(X)$. The $2$-distance graph $T_2(X)$ 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 $T_2(X)=G$. In this paper, we give three characterizations of $2$-distance graphs, and find all graphs $X$ such that $T_2(X)=kP_2$ or $K_m \cup K_n$, where $k \ge 2$ is an integer, $P_2$ is the path of order $2$, and $K_m$ is the complete graph of order $m \ge 1$.

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