Determine whether or not there is a known polynomial$-$time algorithm for solving the problem $($Hamilton Cycle, Travelling salesman problem, Vertex cover, Independent Set, Finding Cliques, Subset Sum, OR Dominating Set$).$ If there is a known polynomial$-$time algorithm for solving the problem, what it it$?$

You must justify why there is no known poly$-$time algorithm OR identify a poly$-$time procedure that solves the problem.

A graph G has n vertices and m edges. The problem is to determine if G contains a cycle of size $3.$

A graph G has n vertices and m edges. The problem is to determine if G contains a cycle.

A graph G has n vertices and m edges. The problem is to determine if G contains a cycle of size at least k$.$