Extension of polynomial-time solvability from H-free graphs to probe H-free graphs

Determine whether every NP-complete problem that is polynomial-time solvable on H-free graphs for a fixed graph H is also polynomial-time solvable on partitioned probe H-free graphs.

Background

Every H-free graph is also a partitioned probe H-free graph by taking all vertices as probes. Consequently, hardness results for H-free graphs transfer immediately to the partitioned probe setting, but polynomial-time algorithms need not transfer because unknown edges among non-probes may introduce additional complexity.

The paper investigates this general question through Colouring and k-Colouring, showing that the answer can be negative for particular forbidden graphs while leaving the general transfer question unresolved.

References

However, it also gives rise to the following research question:

If an \NP-complete problem $\Pi$ is polynomial-time solvable on the class of $H$-free graphs for some graph~$H$, is $\Pi$ also polynomial-time solvable on (partitioned) probe $H$-free graphs?

— Colouring Probe $H$-Free Graphs  (2505.20784 - Paulusma et al., 27 May 2025) in Section 1, Introduction, subsection “Probe H-Free Graphs”