The class $NP$ can be defined by the means of Monadic Second-Order logic going back to Fagin and Feder-Vardi, and also by forbidden expanded substructures (cf. lifts and shadows of Kun and Ne\v{s}et\v{r}il). Consequently, for such problems there is no dichotomy, unlike for $CSP$'s. We prove that ordering problems for graphs defined by finitely many forbidden ordered subgraphs still capture the class $NP$. In particular, we refute a conjecture of Hell, Mohar and Rafiey that dichotomy holds for this class. On the positive side, we confirm the conjecture of Duffus, Ginn and R\"odl that ordering problems defined by one single biconnected ordered graph are $NP$-complete but for the ordered complete graph. An interesting feature appeared and was noticed several times. For finite sets of biconnected patterns (which may be colored structures or ordered structures) complexity dichotomy holds. A principal tool for obtaining this result is known as the Sparse Incomparability Lemma, a classical result in the theory of homomorphisms of graphs and structures. We prove it here in the setting of ordered graphs as a Temporal Sparse Incomparability Lemma for orderings. Interestingly, our proof involves the Lov\'asz Local Lemma.

翻译：类$NP$可通过Fagin与Feder-Vardi提出的单子二阶逻辑定义，亦可通过禁止扩展子结构定义（参见Kun与Nešetřil提出的提升与投影）。因此，此类问题不存在二分性，这与$CSP$问题不同。我们证明了由有限个禁止有序子图定义的图排序问题仍能刻画$NP$类。特别地，我们否定了Hell、Mohar与Rafiey关于该类问题具有二分性的猜想。在积极方面，我们证实了Duffus、Ginn与Rödl的猜想：由单个双连通有序图定义的排序问题（除有序完全图外）均是$NP$完全的。一个有趣的现象被多次注意到：对于双连通模式（可以是着色结构或有序结构）的有限集合，复杂度二分性成立。获得该结果的主要工具是稀疏不可比性引理——图与结构同态理论中的经典结论。我们在有序图背景下将其证明为排序问题的时序稀疏不可比性引理。值得注意的是，我们的证明涉及Lovász局部引理。