Characterizing $2$-Distance Graphs and Solving the Equations $T_2(X)=k{P}_2$ or $K_m\cup K_n$

Ramuel P. Ching, I. J. L. Garces

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 o…