The dimension of a partial order P is the minimum number of linear orders whose intersection is P . There are efficient algorithms to test if a partial order has dimension at most 2. In 1982 Yannakakis [25] showed that for k ≥ 3 to test if a partial order has dimension ≤ k is NP-complete. The height of a partial order P is the maximum size of a chain in P . Yannakakis also showed that for k ≥ 4 to test if a partial order of height 2 has dimension ≤ k is NP-complete. The complexity of deciding whether an order of height 2 has dimension 3 was left open. This question became one of the best known open problems in dimension theory for partial orders. We show that the problem is NP-complete.Technically, we show that the decision problem (3DH2) for dimension is equivalent to deciding for the existence of bipartite triangle containment representations (BTCon). This problem then allows a reduction from a class of planar satisfiability problems (P-3-CON-3-SAT(4)) which is known to be NP-hard.Mathematics Subject Classifications (2010) 06A07, 68Q25, 05C62,

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We explore what could make recognition of particular intersection-defined classes hard. We focus mainly on unit grid intersection graphs (UGIGs), i.e., intersection graphs of unit-length axis-aligned segments and grid intersection graphs (GIGs, which are defined like UGIGs without unit-length restriction) and string graphs, intersection graphs of arc-connected curves in a plane.We show that the explored graph classes are NP-hard to recognized even when restricted on graphs with arbitrarily large girth, i.e., length of a shortest cycle. As well, we show that the recognition of these classes remains hard even for graphs with restricted degree (4, 5 and 8 depending on a particular class). For UGIGs we present structural results on the size of a possible representation, too.

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