Consider a following NP-problem DOUBLE CLIQUE (abbr.: CLIQ$_{2}$): Given a natural number $k>2$ and a pair of two disjoint subgraphs of a fixed graph $G$ decide whether each subgraph in question contains a $k$-clique. I prove that CLIQ$_{2}$ can't be solved in polynomial time by a deterministic TM, which infers $\mathbf{P}\neq \mathbf{NP}$. This proof upgrades the well-known proof of polynomial unsolvability of the partial result with respect to analogous monotone problem CLIQUE (abbr.: CLIQ) as well as my previous presentation that used appropriate 3-value semantics. Note that problem CLIQ$_{2}$ is not monotone and appears more complex than just iterated CLIQ, as the required subgraphs are mutually dependent.

翻译：考虑如下NP问题——双团问题（简称CLIQ$_{2}$）：给定一个自然数$k>2$以及一个固定图$G$的两个不相交子图，判定每个子图是否包含一个$k$-团。本文证明CLIQ$_{2}$无法由确定性图灵机在多项式时间内求解，由此推得$\mathbf{P}\neq \mathbf{NP}$。该证明改进了关于类似单调问题团问题（简称CLIQ）部分结果的多项式不可解性的著名证明，亦改进了本人先前采用适当三值语义的论述。需注意，CLIQ$_{2}$问题并非单调问题，且由于所要求的子图相互依赖，其复杂度高于单纯的迭代CLIQ问题。