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Colouring Probe H-Free Graphs

Authors Daniël Paulusma , Johannes Rauch , Erik Jan van Leeuwen

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LIPIcs.STACS.2026.73.pdf
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ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Problems, reductions and completeness
Keywords
  • colouring
  • probe graph
  • forbidden induced subgraph
  • complexity dichotomy

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Abstract

The NP-complete problems Colouring and k-Colouring (k ≥ 3) are well studied on H-free graphs, i.e., graphs that do not contain some fixed graph H as an induced subgraph. We research to what extent the known polynomial-time algorithms for H-free graphs can be generalized if we only know some of the edges of the input graph. We do this by considering the classical probe graph model introduced in the early nineties. For a graph H, a partitioned probe H-free graph (G,P,N) consists of a graph G = (V,E), together with a set P ⊆ V of probes and an independent set N = V ⧵ P of non-probes, such that G+F is H-free for some edge set F ⊆ binom(N,2). We show the following:  
- We fully classify Colouring on partitioned probe H-free graphs and show that the obtained complexity dichotomy differs from the known dichotomy of Colouring for H-free graphs. 
- We fully classify 3-Colouring on partitioned probe P_t-free graphs: we prove polynomial-time solvability for t ≤ 5 and NP-completeness for t ≥ 6. In contrast, 3-Colouring on P_t-free graphs is known to be polynomial-time solvable for t ≤ 7 and quasi-polynomial-time solvable for t ≥ 8.  Our main result is our polynomial-time algorithm for 3-Colouring on partitioned P₅-free graphs. For this result, and also for all our other polynomial-time results, we do not need to know the edge set F; we only need to know its existence. Moreover, the class of probe P₅-free graphs includes not only paths of arbitrary length but even all bipartite graphs and is much richer than the class of P₅-free graphs. The latter is also evidenced by the fact that there exist graph problems, such as Matching Cut, that are known to be polynomial-time solvable for P₅-free graphs but NP-complete for partitioned probe P₅-free graphs. In particular, unlike the class of 3-colourable P₅-free graphs, the class of 3-colourable probe P₅-free graphs has unbounded mim-width. Hence, our polynomial-time result for 3-Colouring for probe P₅-free graphs suggests that there may be another, deeper overarching reason why 3-Colouring is polynomial-time solvable for P₅-free graphs.

Cite As Get BibTex

Daniël Paulusma, Johannes Rauch, and Erik Jan van Leeuwen. Colouring Probe H-Free Graphs. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 73:1-73:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.STACS.2026.73

Author Details

Daniël Paulusma
  • Department of Computer Science, Durham University, UK
Johannes Rauch
  • Institute of Optimization and Operations Research, Ulm University, Germany
Erik Jan van Leeuwen
  • Department of Information and Computing Sciences, Utrecht University, The Netherlands

Funding

  • Rauch, Johannes: Supported by the German Academic Scholarship Foundation (Studienstiftung des Deutschen Volkes).

Acknowledgements

We thank Clément Dallard and Andrea Munaro for helpful discussions and ideas that ultimately led to the proof of Theorem 3.

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