Colouring of Graphs with Ramsey-Type Forbidden Subgraphs

  • Konrad K. Dabrowski
  • Petr A. Golovach
  • Daniël Paulusma
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8165)


A colouring of a graph G = (V,E) is a mapping c: V → {1,2,…} such that c(u) ≠ c(v) if uv ∈ E; if |c(V)| ≤ k then c is a k-colouring. The Colouring problem is that of testing whether a given graph has a k-colouring for some given integer k. If a graph contains no induced subgraph isomorphic to any graph in some family \({\cal H}\), then it is called \({\cal H}\)-free. The complexity of Colouring for \({\cal H}\)-free graphs with \(|{\cal H}|=1\) has been completely classified. When \(|{\cal H}|=2\), the classification is still wide open, although many partial results are known. We continue this line of research and forbid induced subgraphs {H 1,H 2}, where we allow H 1 to have a single edge and H 2 to have a single non-edge. Instead of showing only polynomial-time solvability, we prove that Colouring on such graphs is fixed-parameter tractable when parameterized by |H 1| + |H 2|. As a byproduct, we obtain the same result both for the problem of determining a maximum independent set and for the problem of determining a maximum clique.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Konrad K. Dabrowski
    • 1
  • Petr A. Golovach
    • 2
  • Daniël Paulusma
    • 3
  1. 1.ESSEC Business SchoolCergy PontoiseFrance
  2. 2.Department of InformaticsBergen UniversityBergenNorway
  3. 3.School of Engineering and Computing Sciences, Science LaboratoriesDurham UniversityDurhamUnited Kingdom

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