Robust linear-time recognition of a dominating pair in AT-free graphs

Develop a robust linear-time algorithm that, given an arbitrary graph, either outputs a dominating pair or correctly reports that the graph is not AT-free.

Background

Every AT-free graph contains a dominating pair, and such a pair can be found in linear time when the input is guaranteed to be AT-free. In contrast, the paper explains that neither recognizing AT-free graphs nor finding a dominating pair in an unrestricted graph admits a linear-time algorithm under the stated fine-grained assumptions. The unresolved issue is therefore whether this guarantee can be made robust: the algorithm must work on arbitrary inputs and distinguish non-AT-free graphs while retaining linear running time.

References

This leads to an interesting open question: is there a robust, linear-time algorithm for finding a dominating pair in an AT-free graph? Here, ``robust'' is used in the sense of ; a robust linear-time algorithm must take an arbitrary (i.e., not necessarily AT-free) graph as input and, in linear time, either output a dominating pair or correctly report that the graph is not AT-free.

— Asteroidal Triples and Friends in (Likely Necessary) Matrix Multiplication Time  (2610.22133 - Nogler et al., 15 Oct 2026) in Section "Open Problems" (following Section 3, "Listing Dominating Pairs in Matrix Multiplication Time")