Abstract
Daligault, Rao and Thomassé asked whether a hereditary class of graphs well-quasi-ordered by the induced subgraph relation has bounded clique-width. Lozin, Razgon and Zamaraev recently showed that this is not true for classes defined by infinitely many forbidden induced subgraphs. However, in the case of finitely many forbidden induced subgraphs the question remains open and we conjecture that in this case the answer is positive. The conjecture is known to hold for classes of graphs defined by a single forbidden induced subgraph H, as such graphs are well-quasi-ordered and are of bounded clique-width if and only if H is an induced subgraph of P4. For bigenic classes of graphs, i.e. ones defined by two forbidden induced subgraphs, there are several open cases in both classifications. In the present paper we obtain a number of new results on well-quasi-orderability of bigenic classes, each of which supports the conjecture.
Highlights
Well-quasi-ordering is a highly desirable property and frequently discovered concept in mathematics and theoretical computer science [18, 22]
One of the first steps towards this result was the proof of the fact that graph classes of bounded treewidth are well-quasi-ordered by the minor relation [26] (a graph parameter π is said to be bounded for some graph class G if there exists a constant c such that π(G) ≤ c for all G ∈ G)
There are six classes of (H1, H2)-free graphs for which we do not know whether they are well-quasi-ordered by the induced subgraph relation, and there is one open case left for the verification of Conjecture 1 for bigenic classes
Summary
Well-quasi-ordering is a highly desirable property and frequently discovered concept in mathematics and theoretical computer science [18, 22]. This was stated as an open problem by Daligault, Rao and Thomasse [15] and a negative answer to this question was recently given by Lozin, Razgon and Zamaraev [23] The latter authors disproved the conjecture by giving a hereditary class of graphs whose set of minimal forbidden induced subgraphs is infinite. Should Conjecture 1 be true, for finitely defined classes of graphs the aforementioned algorithmic consequences of having bounded clique-width hold for the property of being well-quasi-ordered by the induced subgraph relation. In this case, the two notions even coincide: a class of graphs defined by a single forbidden induced subgraph H is well-quasi-ordered if and only if it has bounded clique-width if and only if H is an induced subgraph of P4 (see, for instance, [14, 16, 21]). Six of the nine open cases have bounded clique-width (and immediately verify Conjecture 1), leaving three remaining open cases of bigenic classes for which we still need to verify Conjecture 1
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