Abstract
It is known that the set of all simple graphs is not well-quasi-ordered by the induced subgraph relation, i.e. it contains infinite antichains (sets of incomparable elements) with respect to this relation. However, some particular graph classes are well-quasi-ordered by induced subgraphs. Moreover, some of them are well-quasi-ordered by a stronger relation called labelled induced subgraphs. In this paper, we conjecture that a hereditary class X which is well-quasi-ordered by the induced subgraph relation is also well-quasi-ordered by the labelled induced subgraph relation if and only if X is defined by finitely many minimal forbidden induced subgraphs. We verify this conjecture for a variety of hereditary classes that are known to be well-quasi-ordered by induced subgraphs and prove a number of new results supporting the conjecture.
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Atminas, A., Lozin, V.V. Labelled Induced Subgraphs and Well-Quasi-Ordering. Order 32, 313–328 (2015). https://doi.org/10.1007/s11083-014-9333-9
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DOI: https://doi.org/10.1007/s11083-014-9333-9