I have a problem (A) on undirected graphs that I wish to show is NP-Hard. I can show that there is a reduction from a well known NP-Hard problem (B) to A by constructing an instance of A with a complete graph that solves B.
Since undirected graphs generalize complete graphs, am I safe in saying that A is NP-Hard for undirected graphs? Something about that doesn't seem right to me, since problem A on complete graphs is essentially the "non-graphical" version of the problem. Am I missing something?
Disclaimer: I am new to proving hardness results.