(1)I am just wondering if I have an algorithm to solve HP and are allowed to use it as a subroutine to solve HC. Then if given an input G, can I modify it and apply the HP subroutine and repeat this process in order to answer whether or not G has a HC. Also, the size of modified graph(s) that I input into the HP subroutine and number of repetitions must be bounded by a polynomial in the size of the input graph.
(2)I am just curious about if I can show HP is NP-hard, can I show that HC is also NP-hard.