We study several properties of sets that are complete for NP.
We prove that if $L$ is an NP-complete set and $S \not\supseteq L$ is a p-selective sparse set, then $L - S$ is many-one-hard for NP. We demonstrate existence of a sparse set $S \in \mathrm{DTIME}(2^{2^{n}})$
such that for every $L \in NP - P$, $L - S$ is not many-one-hard for NP.
Moreover, we prove for every $L \in NP - P$, that there exists a
sparse $S \in EXP$ such that $L - S$ is not many-one-hard for NP.
Hence, removing sparse information in P from a complete set leaves the
set complete, while removing sparse information in EXP from a complete
set may destroy its completeness. Previously, these properties were known only for exponential time complexity classes.

We use hypotheses about pseudorandom generators and secure one-way permutations to resolve longstanding open questions about whether NP-complete sets are immune. For example, assuming that pseudorandom generators and secure one-way permutations exist,
it follows easily that NP-complete sets are not p-immune.
Assuming only that secure one-way permutations exist, we prove that
no NP-complete set is $DTIME(2^{n^\epsilon})$-immune. Also, using these
hypotheses we show that no NP-complete set is quasipolynomial-close to P.

We introduce a strong but reasonable hypothesis and infer from it that
disjoint Turing-complete sets for NP are not closed under union. Our hypothesis
asserts existence of a UP-machine $M$ that accepts $0^*$ such
that for some $0 < \epsilon < 1$, no $2^{n^{\epsilon}}$ time-bounded
machine can correctly compute infinitely many accepting
computations of $M$. We show that if
$UP \cap coUP$ contains $DTIME(2^{n^{\epsilon}})$-bi-immune sets, then
this hypothesis is true.