Extending Partial Representations of Circular-Arc Graphs

TL;DR

This paper investigates the complexity of extending partial representations of circular-arc graphs, revealing polynomial algorithms for certain subclasses and NP-completeness for others, highlighting nuanced complexity differences.

Contribution

It demonstrates that the partial representation extension problem is NP-complete for circular-arc graphs but tractable for specific subclasses like Helly circular-arc graphs, with new algorithms provided.

Findings

Recognition of circular-arc graphs is polynomial, but extension is NP-complete.

Linear-time algorithms for extending certain Helly circular-arc representations.

NP-completeness of extension problem for unit circular-arc graphs.

Abstract

The partial representation extension problem generalizes the recognition problem for classes of graphs defined in terms of vertex representations. We exhibit circular-arc graphs as the first example of a graph class where the recognition is polynomially solvable while the representation extension problem is NP-complete. In this setting, several arcs are predrawn and we ask whether this partial representation can be completed. We complement this hardness argument with tractability results of the representation extension problem on various subclasses of circular-arc graphs, most notably on all variants of Helly circular-arc graphs. In particular, we give linear-time algorithms for extending normal proper Helly and proper Helly representations. For normal Helly circular-arc representations we give an $O(n^3)$-time algorithm. Surprisingly, for Helly representations, the complexity hinges on the seemingly irrelevant detail of whether the predrawn arcs have distinct or non-distinct endpoints: In the former case the previous algorithm can be extended, whereas the latter case turns out to be NP-complete. We also prove that representation extension problem of unit circular-arc graphs is NP-complete.