If P=NP, is there a polynomial-time algorithm $A$ that can decide the $\text{Independent Set}$ decision problem?
That is, with an undirected graph $G = (V, E)$ and a positive integer $k$, does $G$ contain a set of $k$ independent vertices (i.e. a set of vertices no two of which are joined by an edge)?
I am unsure of whether to look at $\text{Independent Set for trees}$ or algorithms that could perhaps solve problems equivalent to $\text{Clique, Vertex Cover}$ etc. in polynomial-time. I would appreciate some guidance on my options and the following development...
Upon choosing $A$, how could I use it to create a further polynomial-time algorithm that finds an $\text{Independent Set}$ of size $k$, rather than just returning a Boolean truth value.