Partial and Simultaneous Transitive Orientations via Modular Decomposition

TL;DR

This paper introduces linear-time algorithms for transitive orientation extension and simultaneous representation problems in comparability, permutation, and circular permutation graphs using modular decomposition techniques.

Contribution

It provides the first efficient algorithms for these problems on circular permutation graphs and improves existing algorithms for permutation graphs.

Findings

Linear-time algorithms for transitive orientation extension.

Linear-time algorithms for sunflower orientation problems.

First efficient algorithms for circular permutation graphs.

Abstract

A natural generalization of the recognition problem for a geometric graph class is the problem of extending a representation of a subgraph to a representation of the whole graph. A related problem is to find representations for multiple input graphs that coincide on subgraphs shared by the input graphs. A common restriction is the sunflower case where the shared graph is the same for each pair of input graphs. These problems translate to the setting of comparability graphs where the representations correspond to transitive orientations of their edges. We use modular decompositions to improve the runtime for the orientation extension problem and the sunflower orientation problem to linear time. We apply these results to improve the runtime for the partial representation problem and the sunflower case of the simultaneous representation problem for permutation graphs to linear time. We also give the first efficient algorithms for these problems on circular permutation graphs.