A characterization of maximal 2-dimensional subgraphs of transitive graphs

A transitive graph is 2-dimensional if it can be represented as the intersection of two linear orders. Such representations make answering of reachability queries trivial, and allow many problems that are NP-hard on arbitrary graphs to be solved in polynomial time. One may therefore be interested in finding 2-dimensional graphs that closely approximate a given graph of arbitrary order dimension. In this paper we show that the maximal 2-dimensional subgraphs of a transitive graph G are induced by the optimal near-transitive orientations of the complement of G. The same characterization holds for the maximal permutation subgraphs of a transitively orientable graph. We provide an algorithm that enables this problem reduction in near-linear time, and an approach for enlarging non-maximal 2-dimensional subgraphs, such as trees.