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The following three theorems explore well-quasi-ordering of the classes above the Bell number with finite distinguishing number.

Theorem 5.5.4. Suppose a class X with finite distinguishing number kX is above

the Bell number. Then X is not well-quasi-ordered by labelled induced subgraph

relation.

Proof. From Theorem 5.3.5 we obtain that X contains P(w, H) with w =a1a2. . .

almost periodic andH prime. LetM be the smallest number such that each letter of w appears at least 5×2|H| times. It is possible to pick such M because w is almost periodic. For i ≥ M, let Gi = Gw,H(1,2, . . . , i). Then Gi is a strong

(2,|H|)-graph with its letter partition and φ(Gi) =Pi - a path with i vertices. By

Lemma 5.2.4 all the embeddingsGi→Gj must correspond to embeddingsPi →Pj

and must respect the labels of the vertices. However, the paths form an antichain with respect to labelled induced subgraph relation. This implies that graphs Gi

form an antichain with respect to labelled induced subgraph relation as well. SoX

Theorem 5.5.5. Suppose a class X = F ree(F1, F2, . . . , Fn) is above Bell num-

ber and has finite distinguishing number. Then X is not well-quasi-ordered by the

induced subgraph relation.

Proof. LetX be as in the statement. By Lemma 5.4.2 (c) we obtain thatX contains

a periodic classP(w, H) with some w= (a1a2. . . ak)∞ and we can assume that H

is prime. Letm=max(|F1|,|F2|, . . . ,|Fn|) and let M =max(5×2k+1, m+ 1). For

each i≥ M, let Gi be a graph obtained from Gw,H(1,2, . . . , ik) by swapping edge

with non-edge, between vertices 1 andik. More formally, letGibe a graph on vertex

set [ik] with vertices j < j0 adjacent to each other if and only if (j, j0)6= (1, k) and jj0 ∈E(Gw,H(1,2, . . . , ik) or (j, j0) = (1, ik) and jj0 ∈/E(Gw,H(1,2, . . . , ik).

Notice that each Gi is a strong (|H|,2) graph, with |H| bags in the prime

partition, and it’s sparsification φ(Gi) is a cycle Cik. Therefore, by Lemma 5.2.4

the set of graphsS={Gi:i≥M} is an antichain.

We finish the proof by showing that S ⊂ X. Notice that deleting a vertex l, for some 1 ≤ l ≤ ik of Gi we are left with a graph isomorphic to Gw,H(l+

1, l+ 2, . . . , l+ik −1) which is in P(w, H) and hence in X. We deduce that all proper induced subgraphs ofGi are in X. Now for all j, Hj ∈ X/ and |Gi|>|Hj|.

Hence Hj is not an induced subgraph of Gi. As this holds for all j, we haveGi ∈

F ree(F1, F2, . . . , Fn). So for alli≥M,Gi ∈ X, hence S⊂ X.

Theorem 5.5.6. Each minimal class above the Bell number X =P(w, H) withw

almost periodic andH prime is well-quasi-ordered by the induced subgraph relation.

Proof. Suppose we have an antichain S ∈ X. Then for any G ∈ S we have that

S\{G}is an antichain inF ree(G)∩ X. The classF ree(G)∩ X is below Bell number and hence by Theorem 5.5.3, it is well-quasi-ordered. HenceS\{G} is finite which means that S is finite. Hence X is well-quasi-ordered by the induced subgraph relation.

We end this section with conjecture on which hereditary classes (not necesser- ily finitely defined) with finite distinguishing number are well-quasi-ordered by the induced subgraph relation. The conjecture is based on the results we obtained in this section and the result of Guoli Ding for monotone classes (closed under taking subgraphs). In [Ding, 1992] the author proved that a monotone class is well-quasi- ordered if and only if it contains finitely many graphs from two antichains: the set of cycles and the set of so-calledH-graphs. Let us call the antichains produced in the proof of Theorem 5.5.5 the antichains of generalised cycles. Define the antichains of generalised H-graphs in a similar way.

Conjecture 5.5.1. A hereditary class with finite distinguishing number is well- quasi-ordered by the induced subgraph relation if and only if it contains only finitely many elements from every antichain of generalised cycles and generalisedH-graphs.

5.6

Conclusion

In this part, we have characterised all minimal hereditary classes of graphs whose speed is at least the Bell number Bn. This characterisation allowed us to show

that the problem of determining if the speed of a hereditary class X defined by finitely many forbidden induced subgraphs is above or below the Bell number is decidable, i.e., there is an algorithm that gives a solution to this problem in a finite number of steps. However, the complexity of this algorithm, in terms of the input forbidden graphs, remains an open question. In particular, it would be interesting to determine if there is a polynomial bound on the minimum` such that the input classX contains an `-factor as in Theorem 5.4.2(d) if it is above the Bell number, and it fails to contain any (`, `)-strip as in Theorem 5.4.2(e) if it is below.

We also verified the conjecture of Daligault et al. [2010] and proved decidabil- ity of well-quasi-ordering for classes of finite distinguishing number. The boundary of well-quasi-ordering coincided with the boundary separating classes ofk-uniform graphs from the classes for which the uniformicity is unbounded.

To settle the conjecture of Daligault et al. [2010] and to obtain results about deciding well-qausi-ordering for classes with infinite distinguishing number, we sug- gest to study letter graphs which are known to be well-quasi-ordered and try to identify the minimal classes of unbounded lettericity. The classes of unbounded lettericity should contain infinite antichains similar to the ones we obtained in this Chapter and the ones we described in Section 2.6. A further indication that this is a promising direction is due to a certain orthogonality between the notions of uniformicity and lettericity. Indeed, the building blocks ofk-uniform graphs consist of bags which are independent set of cliques with possible matchings or comatchings between them, but a class of matchings has unbounded lettericity. On the other hand, the letter graphs are obtained when instead of matchings and comatchings we use chain graphs, but a class of chain graphs has unbounded uniformicity. Finally, we note that despite the similarity to the question we solved in this section, the question for letter graphs is still a very challenging research question. In particular, adapting our approach to letter graphs, one would need to develop sparsification tools for chain graphs which seems to be a hard task.

Chapter 6

Subquadratic properties

We say that a hereditary graph property X is subquadratic if there is a function f(n) = o(n2_{) bounding the number of edges in all n-vertex graphs in X. The}

family of subquadratic properties contains many important classes such as graphs of bounded vertex degree, of bounded tree-width, all proper minor closed graph classes. In all these examples, the number of edges is bounded by a linear function in the number of vertices and all of the listed properties are rather small (see e.g. [Norine et al., 2006] for the number of graphs in proper minor closed graph classes). In the terminology of [Balogh et al., 2000], they all are at most factorial. The family of subquadratic properties is much wider and contains classes with a superfactorial speed of growth, such as projective plane graphs (or more generally C4-free bipar-

tite graphs), in which case the number of edges is Θ(n32). In fact, as we show in

Section 6.1, subquadratic properties have a nice structural characterization: these are precisely hereditary classes of graphs without large bicliques as subgraphs.

Our main result of this Chapter is proved in Section 6.2: a subquadratic property which is well-quasi-ordered by the induced subgraph relation is of bounded tree-width. Since bounded tree-width implies bounded clique-width, this verifies (and even strengthens) the conjecture of Daligault et al. [2010] for subquadratic properties. In Section 6.3 we further show that for a finitely defined subquadratic property, it is a decidable task to test well-quasi-orderability by the induced sub- graphs relation. As a byproduct, in Section 6.4 we show that computing a biclique of orderkin aPs-free graphs is fixed parameter linear when parameterised bykand

s.

All preliminary information related to the topic of the chapter can be found in Section 6.1, where we also prove a number of preparatory results.