$1.$ Showing that a problem is $$NP$\_$complete$$ requires $$Polynomial$-$time Reduction.$$ Now, a set of

nodes $$V$$ is a vertex cover. The reason it is called a vertex cover is that the removal of $$V$$

from the graph destroys every edge. Finding a vertex cover in a graph is not easy and it is the

VERTEX$\_$COVER problem. In the graph shown below in Fig. $1,\{$A$,$ C$,$ D$,$ F$\}$ is a set of vertex

covers. This is just an example, not related to your answers. Now, given the input as the graph

$$H$$ and an integer k$,$ does there exist a vertex$\_$cover of $$H$$ at most k nodes? Be creative and

explain your answer.

Fig. $1$ An example graph