Complexity of colouring on partitioned probe H-free graphs for unresolved linear forests

Determine the computational complexity of 3-Colouring on partitioned probe H-free graphs when H is one of 2P2+sP1 for s≥1, P3+P2+sP1 for s≥0, P4+sP1 for s≥1, P4+P2+sP1 for s≥0, or P5+sP1 for s≥1.

Background

The paper establishes polynomial-time solvability for 3-Colouring on partitioned probe (P3+sP1)-free graphs and NP-completeness on partitioned probe (P6,3P2,2P3)-free graphs. Together with known results for H-free graphs when H is not a linear forest, these results reduce the unresolved classification to the listed families of disconnected linear forests.

The open cases concern determining, for each specified forbidden induced subgraph H, whether incomplete edge information in the partitioned probe model preserves polynomial-time tractability or causes NP-completeness.

References

Theorem~\ref{thm:3col-probe-p5}, Theorems~\ref{thm:3col-probe-p6}--\ref{thm:3col-probe-sp1p3} and the result that {$3$} is \NP-complete on $H$-free graphs if $H$ is not a linear forest leave only the following open cases:

What is the complexity of {3} on partitioned probe $H$-free graphs when $H$ is $2P_2+sP_1$ ($s \geq 1$), $P_3+P_2+sP_1$ ($s \geq 0$), $P_4+sP_1$ ($s \geq 1$), $P_4+P_2+sP_1$ ($s \geq 0$), or $P_5+sP_1$ ($s \geq 1$)?

— Colouring Probe $H$-Free Graphs  (2505.20784 - Paulusma et al., 27 May 2025) in Section 6, Additional Results and Concluding Remarks