Question 32
For a finite simple graph $G$, we define $G'$ to be the graph on the same vertex set as $G$, where for any two vertices $u \neq v$, the pair $\{u, v\}$ is an edge of $G'$ if and only if $u$ and $v$ have a common neighbor in $G$. Prove that if $G$ is a finite simple graph which is isomorphic to $(G')'$, then $G$ is also isomorphic to $G'$.
We say a vertex of a graph is *fatal* if it has degree at least 3, and some two of its neighbors are not adjacent.
We say a vertex of a graph is *fatal* if it has degree at least 3, and some two of its neighbors are not adjacent.