such that $$a_i<a_j\textnormal{ for each }(a_i,a_j)\in \bigcup_{1 \leq \ell \leq N} B_\ell,$$ and $$a_i<a_j<a_k \textnormal{ or } a_j<a_k<a_i\textnormal{ for each }(a_i,a_j,a_k)\in \bigcup_{1 \leq \ell \leq N} C_\ell\rm{?}$$\
Let $I$ be an instance of [$N$-Disjoint-Intermezzo]{}, and let ${{\mathcal O}}$ be an ordering on the elements of $A$ that satisfies the two ordering constraints for each pair and triple in the statement of [$N$-Disjoint-Intermezzo]{}. We say that ${{\mathcal O}}$ is an [*[$N$-Disjoint-Intermezzo]{} ordering*]{} for $I$. We next show that [$4$-Disjoint-Intermezzo]{} is NP-complete via reduction from the following restricted version of [3-SAT]{}.

[2P2N-3-SAT]{}\
[**Instance.**]{} A set $U$ of variables, and a set ${{\mathcal C}}$ of clauses, where each clause is a disjunction of exactly three literals, such that each variable appears negated exactly twice and unnegated exactly twice in ${{\mathcal C}}$.\
[**Question.**]{} Does there exist a truth assignment for $U$ that satisfies each clause in ${{\mathcal C}}$?

Berman et al. [@berman03 Theorem 1] established NP-completeness for [2P2N-3-SAT]{}.

\[t:4-DI\] [4-Disjoint-Intermezzo]{} is [*NP*]{}-complete.

We show that the construction by Guttmann and Maucher [@guttmann06 Lemma 1], that was used to show that [Intermezzo]{} is NP-complete via reduction from [3-SAT]{}, yields an instance of [4-Disjoint-Intermezzo]{} if we reduce from [2P2N-3-SAT]{}. Using the same notation as Guttmann and 