For the purpose of this question, consider a $$vertex rearranging function$$ for a graph G $=($V$,$ E$)$ is

a function $\backslash$phi : V $->$ V that rearranges the vertices of G while preserving the graph$$s structure. This

$1$

means that if there is an edge between vertices u and v in G$,$ then there must also be an edge between

$\backslash$phi $($u$)$ and $\backslash$phi $($v$)$ in G$,$ and if there is no edge between u and v$,$ then there is none between $\backslash$phi $($u$)$ and

$\backslash$phi $($v$).$ In other words, $\backslash$phi is a bijective map from V to V such that

$($u$,$ v$)$ in E

implies $(\backslash$phi $($u$),\backslash$phi $($v$))$ in E and $($u$,$ v$)$ in $/$ E implies $(\backslash$phi $($u$),\backslash$phi $($v$))$ in $/$ E$.$