# A characterization of maximal 2-dimensional subgraphs of transitive graphs

## TL;DR

This paper characterizes maximal 2-dimensional subgraphs of transitive graphs, linking them to optimal near-transitive orientations of the complement, and provides algorithms for their identification and enlargement.

## Contribution

It introduces a novel characterization of maximal 2-dimensional subgraphs via near-transitive orientations and offers efficient algorithms for their detection and expansion.

## Key findings

- Maximal 2-dimensional subgraphs are induced by optimal near-transitive orientations.
- The characterization applies to permutation subgraphs of transitively orientable graphs.
- An algorithm for near-linear time reduction is provided.

## Abstract

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.