Consider the following two problems:

Clique

INPUT: an undirected graph $G=(V,E)$ and a natural number $k,$ where $1k|V|.$

QUESTION: Does $G$ have a $k-$clique? I.e$,$ does there exist $V^{\text{'}}$subeV such that $|V^{\text{'}}|=k$ and every pair of vertices in $V^{\text{'}}$ is connected by an edge in $E?$

QUARTERClique $($QC$)$

INPUT: an undirected graph $G=(V,E).$

QUESTION: Does $G$ have a clique with $|)/(4|$ vertices? I.e$,$ does there exist $V^{\text{'}}$subeV such that $|)/(4|$ and every pair of vertices in $V^{\text{'}}$ is connected by an edge in $E?$

Prove that QUARTER$-$CliQUE is NP$-$complete by constructing a ${?}_m^P-$reduction from $k-$ClIQUE:

Describe $a{?}_m^P-$reduction $f$ from Clique to QC$.$

You should describe how an instance of CliQuE is transformed into an instance of QC$.$ Make sure that your reduction is in the correct direction for showing that QC is NP$-$complete. Pictures are great, but do include enough English to make your description unambiguous. N$.$B$.$: in the definition of Clique, the value $k$ can range from $1$ to $|V|.$

Prove that if $(G_1,k)in$ Clique then $G_2$inQC where $G_2=f(G_1,k)$ for the function $f$ you described in part $1.$

Prove that if $G_2$inQC then $(G_1,k)in$ Clique where $G_2=f(G_1,k)$ for the function $f$ you described in part $1.$

N$.$B$.$: the proofs in parts $2$ and $3$ must be separate proofs. Also, these proofs must hold for all undirected graphs $G_1$ and all values of $k,$ not just conveniently chosen examples. $($That$\text{'}$s why a proof is required.$)$