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2 results

  • Infinite induced-saturated graphs

  • Part of
  • Marthe Bonamy, Carla Groenland, Tom Johnston, Natasha Morrison, Alexander Scott
  • Journal:
    Canadian Journal of Mathematics , First View
    Published online by Cambridge University Press:
    24 March 2026, pp. 1-31
    • Article
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  • A graph G is H-induced-saturated if G is H-free but deleting any edge or adding any edge creates an induced copy of H. There are nontrivial graphs H, such as $P_4$, for which no finite H-induced-saturated graph G exists. We show that for every finite graph H that is not a clique or an independent set, there always exists a countable H-induced-saturated graph. In fact, we show that a far stronger property can be achieved: there is a countably infinite H-free graph G such that any graph $G'\ne G$ obtained by making a locally finite set of changes to G contains a copy of H.

  • 1 - Clique-width for hereditary graph classes

  • Edited by Allan Lo, University of Birmingham, Richard Mycroft, University of Birmingham, Guillem Perarnau, Universitat Politècnica de Catalunya, Barcelona, Andrew Treglown, University of Birmingham
  • Book:
    Surveys in Combinatorics 2019
    Published online:
    17 June 2019
    Print publication:
    27 June 2019, pp 1-56
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  • Summary

    Clique-width is a well-studied graph parameter owing to its use in understanding algorithmic tractability: if the clique-width of a graph class G is bounded by a constant, a wide range of problems that are NP-complete in general can be shown to be polynomial-time solvable on G. For this reason, the boundedness or unboundedness of clique-width has been investigated and determined for many graph classes. We survey these results for hereditary graph classes, which are the graph classes closed under taking induced subgraphs. We then discuss the algorithmic consequences of these results, in particular for the COLOURING and GRAPH ISOMORPHISM problems. We also explain a possible strong connection between results on boundedness of clique-width and on well-quasi-orderability by the induced subgraph relation for hereditary graph classes.