Why is determining the size of a maximum independent set or a clique in P?
I would be interested to see what you read, but determining the size of the maximum independent set is $NP$-hard (of course clique is equivalent, so I'll just talk about independent set). Recall that the decision problem "Does the graph $G$ contain an indepedent set of size at least $k$?" is $NP$-complete. If we could determine the size of the maximum independent set ($\alpha(G)$) in polynomial time, we could answer this decision problem in polynomial time by taking $G$, getting $\alpha(G)$ from our polynomial time algorithm, then if $k \leq \alpha(G)$, we say $Yes$, if $k &gt; \alpha(G)$ we say no. Hence a polynomial time algorithm for determining the maximum would immediately give us $P=NP$.

Perhaps you read that determining the size of a maximal independent set is poynomial time solvable. A maximal set is just one that cannot be made larger, so we can (typically, and certainly for independent sets) compute these greedily.