Well-Quasi-Order for Permutation Graphs Omitting a Path and a Clique

We consider well-quasi-order for classes of permutation graphs which omit both a path and a clique. Our principle result is that the class of permutation graphs omitting $P_5$ and a clique of any size is well-quasi-ordered. This is proved by giving a structural decomposition of the corresponding permutations. We also exhibit three infinite antichains to show that the classes of permutation graphs omitting $\{P_6,K_6\}$, $\{P_7,K_5\}$, and $\{P_8,K_4\}$ are not well-quasi-ordered.


INTRODUCTION
While the Minor Theorem of Robertson and Seymour [17] shows that the set of all graphs is wellquasi-ordered under the minor relation, it is well known that this set is not well-quasi-ordered under the induced subgraph order. Consequently, there has been considerable interest in determining which classes of graphs are well-quasi-ordered under this order. Here we consider finite graphs and permutation graphs which omit both a path P k and a clique K ℓ . Our main result, proved in Section 4, is that permutation graphs which avoid both P 5 and a clique K ℓ are well-quasi-ordered under the induced subgraph order for every finite ℓ. We also prove, in Section 5, that the three classes of permutation graphs defined by forbidding tP 6 , K 6 u, tP 7 , K 5 u and tP 8 , K 4 u respectively are not well-quasi-ordered, by exhibiting an infinite antichain in each case.  Figure 1: This figure shows the known wqo results for classes of graphs and permutation graphs avoiding paths and cliques, including the results of this paper. Filled circles indicate that all graphs avoiding the specified path and clique are wqo. Half-filled circles indicate that the corresponding class of permutation graphs are wqo, but that the corresponding class of all graphs are not wqo. Empty circles indicate that neither class is wqo. Note that for the three unknown cases (indicated by question marks), it is known that the corresponding class of graphs contains an infinite antichain.
contains neither an infinite strictly decreasing sequence nor an infinite antichain (a set of pairwise incomparable graphs) under the induced subgraph ordering. Note that as we are interested only in classes of finite graphs, wqo is synonymous with a lack of infinite antichains.
Given a permutation π " πp1q¨¨¨πpnq, its corresponding permutation graph is the graph G π on the vertices t1, . . . , nu in which i is adjacent to j if both i ď j and πpiq ě πpjq. This mapping is many-to-one, because, for example, G 231 -G 312 -P 3 . Given permutations σ " σp1q¨¨¨σpkq and π " πp1q¨¨¨πpnq, we say that σ is contained in π and write σ ď π if there are indices 1 ď i 1 ă¨¨¨ă i k ď n such that the sequence πpi 1 q¨¨¨πpi k q is in the same relative order as σ. For us, a class of permutations is a set of permutation closed downward under this containment order, and a class is wqo if it does not contain an infinite antichain.
The mapping π Þ Ñ G π is easily seen to be order-preserving, i.e., if σ ď π then G σ is an induced subgraph of G π . Therefore if a class C of permutations is wqo then the associated class of permutation graphs tG π : π P Cu must also be wqo. However, there are infinite antichains of permutations which do not correspond to antichains of permutation graphs. For this reason, when showing that classes of permutation graphs are wqo we instead prove the stronger result that the associated permutation classes are wqo, but when constructing infinite antichains, we must construct antichains of permutation graphs.
A summary of our results is shown in Figure 1, with our new contributions in the upper-right highlighted. The rest of this paper is organised as follows: in Section 2 we briefly summarise the status of the analogous question for non-permutation graphs. Section 3 sets up the necessary notions from the study of permutation classes, before the proof of the well-quasi-orderability of P 5 , K ℓ -free permutation graphs in Section 4. Section 5 contains three non-wqo results, Section 6 briefly presents some enumerative consequences of our results, and the final section contains a few concluding remarks about the three remaining open cases.

NON-PERMUTATION GRAPHS
When the graphs needn't be permutation graphs, well-quasi-ordering is of course harder to attain. One the side of wqo, graphs avoiding K 2 are trivially wqo and graphs avoiding P 4 (co-graphs) are well-known to be wqo, so there are only two nontrivial results, namely graphs avoiding K 3 and P 5 or P 6 . Of course, it suffices to show that K 3 , P 6 -free graphs are wqo, and this was recently proved by Atminas and Lozin [6].
On the side of non-wqo, two infinite antichains are required: one (from [13]) in the class of graphs with neither P 5 (or even 2K 2 ) nor K 4 , and one (from [14]) in the class omitting both P 7 and K 3 . For completeness, we outline both constructions here.
For graphs which omit P 5 and K 4 , Korpelainen and Lozin [13] construct an infinite antichain by adapting a correspondence between permutations and graphs due to Ding [8]. The correspondence we require for our antichain can be described as follows, and is accompanied by Figure 2. For a permutation π of length n, first note that π can be thought of as a structure with n points, equipped with two linear orderings.
With this in mind, form a graph B π which consists of three independent sets, U , V and W , each containing n vertices. Let U " tu 1 , u 2 , . . . , u n u. Between U and V , there is a chain graph: vertex u i in U is adjacent to i vertices of V , and for i ą 1 the neighbourhood of u i´1 in V is contained in the neighbourhood of u i in V . Note that this containment of neighbourhoods defines a linear ordering on the vertices of U : u 1 ă u 2 ă¨¨¨ă u n .
Next, between U and W , we build another chain graph. This time, vertex u πpiq has i neighbours in W , and for i ą 1 the neighbourhood of u πpiq in W contains the neighbourhood of u πpi´1q in W . This defines a second linear ordering on U , namely u πp1q ă u πp1q ă¨¨¨ă u πpnq , and hence π has been encoded in B π . Finally, to complete the construction, between V and W there is a complete bipartite graph, i.e. every edge is present.
Now it is routine to verify that B π does not contain 2K 2 ď P 5 or K 4 for any π. Moreover for permutations σ and π, we have σ ď π if and only if B σ ď B π as induced subgraphs. Thus, one may take any infinite antichain of permutations (for example, the "increasing oscillating" antichain), and encode each element of the antichain as a graph, yielding an infinite antichain in the class of 2K 2 , K 4 -free graphs.
For graphs which omit P 7 and K 3 , a modification of this construction was used by [14] to answer a question of Ding [8], who had asked whether P 7 -free bipartite graphs are wqo. Starting with B π , the modification "splits" every vertex u of U into two, u p1q and u p2q , with an edge between them.
The new vertex u p1q takes the neighbourhood of u with V , while u p2q takes the neighbourhood of u with W . The result is a graph with four independent sets U p1q , U p2q , V and W each of size n, with a perfect matching between U p1q and U p2q , a complete bipartite graph between V and W , and chain graphs between U p1q and V , and between U p2q and W .
This construction yields a 2P 3 -free bipartite graph. Moreover, the permutation π is still encoded in such a way as to ensure an infinite antichain of permutations maps to an infinite antichain of graphs.

STRUCTURAL TOOLS
Instead of working directly with permutation graphs, we establish our wqo results for the corresponding permutation classes (which, by our observations in the introduction, is a stronger result). The permutations 24153 and 31524 are the only permutations which correspond to the permutation graph P 5 , and thus to establish our main result we must determine the structure of the permutation class Avp24153, 31524, ℓ¨¨¨21q.
Considering the wqo problem from the viewpoint of permutations has the added benefit of allowing us to make use of the recently developed tools in this field. In particular, we utilise grid classes and the substitution decomposition. Thus before establishing our main result we must first introduce these concepts.
We frequently identify a permutation π of length n with its plot, the set tpi, πpiqq : 1 ď i ď nu of points in the plane. We say that a rectangle in the plane is axis-parallel if its top and bottom sides are parallel with the x-axis while its left and right sides are parallel with the y-axis. Given natural numbers i and j we denote by ri, js the closed interval ti, i`1, . . . , ju and by ri, jq the half-open interval ti, i`1, . . . , j´1u. Thus the axis-parallel rectangles we are interested in may be described by rx, x 1 sˆry, y 1 s for natural numbers x, x 1 , y, and y 1 .
Monotone grid classes are a way of partitioning the entries of a permutation (or rather, its plot) into monotone axis-parallel rectangles in a manner specified by a 0{˘1 matrix. In order for these matrices to align with plots of permutations, we index them with Cartesian coordinates. Suppose that M is a tˆu matrix (thus M has t columns and u rows). An M -gridding of the permutation π of length n consists of a pair of sequences 1 " c 1 ď¨¨¨ď c t`1 " n`1 and 1 " r 1 ď¨¨¨ď r u`1 " n`1 such that for all k and ℓ, the entries of (the plot of) π that lie in the axis-parallel rectangle rc k , c k`1 qˆrr ℓ , r ℓ`1 q are increasing if M k,ℓ " 1, decreasing if M k,ℓ "´1, or empty if M k,ℓ " 0.
We say that the permutation π is M -griddable if it possesses an M -gridding, and the grid class of M , denoted by GridpM q, consists of the set of M -griddable permutations. We further say that the permutation class C is M -griddable if C Ď GridpM q, and that this class is monotone griddable if there is a finite matrix M for which it is M -griddable.
Grid classes were first described in this generality (albeit under a different name) by Murphy and Vatter [16], who studied their wqo properties. To describe their result we need the notion of the cell graph of a matrix M . This graph has vertex set tpi, jq : M i,j ‰ 0u and pi, jq is adjacent to pk, ℓq if they lie in the same row or column and there are no nonzero entries lying between them in this row or column. We typically attribute properties of the cell graph of M to M itself; thus we say that M is a forest if its cell graph is a forest. (This is a slightly different form of the result than is stated in [16], but the two forms are equivalent as shown by Vatter and Waton [19], who also gave a much simpler proof of Theorem 3.1.) The monotone griddable classes were characterised by Huczynska and Vatter [12]. In order to present this result we also need some notation. Given permutations σ and τ of respective lengths m and n, their sum is the permutation π ' τ whose plot consists of the plot of τ above and to the right of the plot of σ. More formally, this permutation is defined by The obvious symmetry of this operation (in which the plot of τ lies below and to the right of the plot of σ) is called the skew sum of σ and τ and is denoted σ a τ . We can now state the characterisation of monotone griddable permutation classes. The reader might note that the classes we are interested in are not monotone griddable, let alone M -griddable for a forest M . However, our proof will show that these classes can be built from a monotone griddable class via the "substitution decomposition", which we define shortly. Before this, though, we must introduce a few more concepts concerning monotone grid classes, the first two of which are alternative characterisations of monotone griddable classes.
Given a permutation π, we say that the axis-parallel rectangle R is monotone if the entries of π which lie in R are monotone increasing or decreasing (otherwise R is non-monotone). We say that the permutation π can be covered by s monotone rectangles if there is a collection R of s monotone axis-parallel rectangles such that every point in the plot of π lies in at least one rectangle in R. Clearly if C Ď GridpM q for a tˆu matrix M then every permutation in C can be covered by tu monotone rectangles. To see the other direction, note that every permutation which can be covered by s monotone rectangles is M -griddable for some matrix M of size at most p2s´1qˆp2s´1q. There are only finitely many such matrices, say M p1q , . . . , M pmq , so their direct sum, ‚ is a finite matrix whose grid class contains all such permutations. } This characterisation of monotone griddability is recorded in Proposition 3.3 below, which also includes a third characterisation. We say that the line L slices the rectangle R if L X R ‰ H. If C Ď GridpM q for a tˆu matrix then for every permutation π P C there is a collection of t`u horizontal and vertical lines (the grid lines) which slice every non-monotone axis-parallel rectangle of π. Conversely, every such collection of lines defines a gridding of π, completing the sketch of the proof of the following result.
} By adapting this argument it follows that if every permutation in the class C lies in the grid class of a forest of size at most tˆu, then C itself lies in the grid class of a (possibly much larger) forest.

Proposition 3.3 (specialising Vatter [18, Proposition 2.3]).
For a permutation class C, the following are equivalent: (1) C is monotone griddable, (2) there is a constant ℓ such that for every permutation π P C the set of non-monotone axis-parallel rectangles of π can be sliced by a collection of ℓ horizontal and vertical lines, and (3) there is a constant s such that every permutation in C can be covered by s monotone rectangles.
We now move on to the substitution decomposition, which will allow us to build the classes we are interested in from grid classes of forests. An interval in the permutation π is a set of contiguous indices I " ra, bs such that the set of values πpIq " tπpiq : i P Iu is also contiguous. Given a permutation σ of length m and nonempty permutations α 1 , . . . , α m , the inflation of σ by α 1 , . . . , α m -denoted σrα 1 , . . . , α m s -is the permutation of length |α 1 |`¨¨¨`|α m | obtained by replacing each entry σpiq by an interval that is order isomorphic to α i in such a way that the intervals themselves are order isomorphic to σ. Thus the sum and skew sum operations are particular cases of inflations: σ ' τ " 12rσ, τ s and σ a τ " 21rσ, τ s. Given two classes C and U, the inflation of C by U is defined as CrUs " tσrα 1 , . . . , α m s : σ P C m and α 1 , . . . , α m P Uu.
The class C is said to be substitution closed if CrCs " C. The substitution closure, xCy, of a class C is defined as the smallest substitution closed class containing C. A standard argument shows that xCy exists, and by specialising a result of [4] we obtain the following.

Theorem 3.4 (Albert, Ruškuc, and Vatter [4, specialisation of Theorem 4.4]). If the matrix M is a forest then the class xGridpM qy is wqo.
The permutation π is said to be simple if it cannot be written as a nontrivial inflation. Thus if π has length n, it is simple if and only if its only intervals have length 0, 1, and n. Thus every permutation can be expressed as the inflation of a simple permutation. Moreover, in most cases, this decomposition is unique. A permutation is said to be sum (resp., skew) decomposable if it can be expressed as the sum (resp., skew sum) of two shorter permutations. Otherwise it is said to be sum (resp., skew) indecomposable. [1]). Every permutation π except 1 is the inflation of a unique simple permutation σ. Moreover, if π " σrα 1 , . . . , α m s for a simple permutation σ of length m ě 4, then each interval α i is unique. If π is an inflation of 12 (i.e., is sum decomposable), then there is a unique sum indecomposable α 1 such that π " α 1 ' α 2 . The same holds, mutatis mutandis, with 12 replaced by 21 and sum replaced by skew.

Proposition 3.5 (Albert and Atkinson
We close this section by noting how easily this machinery can show that permutation graphs omitting both P k and K 3 are wqo for all k, a result originally due to Korpelainen and Lozin [14]. The corresponding permutations all avoid 321 and thus lie in the grid class of an infinite matrix, known as the infinite staircase (see Albert and Vatter [5]). Moreover, the sum indecomposable permutations which avoid 321 and the two permutations corresponding to P k can be shown to lie in the grid class of a finite staircase,¨1 1 Finally, it follows by an easy application of Higman's Theorem [11] that if the sum indecomposable permutations in a class are wqo, then the class itself is wqo. (In this case, the wqo conclusion also follows by Theorem 3.4.)

PERMUTATION GRAPHS OMITTING P 5 AND K ℓ
In this section we prove that the class of permutations corresponding to permutation graphs omitting P 5 and K ℓ , Avp24153, 31524, ℓ¨¨¨21q, is wqo. Our proof basically consists of two steps. First, we show that the simple permutations in these classes are monotone griddable, and then we show that these griddings can be refined to forests. The conclusion then follows from Theorem 3.4.
Given a set of points in the plane, their rectangular hull is defined to be the smallest axis-parallel rectangle containing all of them. We begin with a very simple observation about these simple permutations.
Proposition 4.1. For every simple permutation π P Avp24153, 31524q, either its greatest entry lies to the left of its least entry, or its leftmost entry lies above its rightmost entry.
Proof. Suppose, for a contradiction, that π is a simple permutation in Avp24153, 31524q such that its greatest entry lies to the right of its least, and its leftmost entry lies below its rightmost entry. Thus, these four extremal entries form the pattern 2143, and the situation is as depicted in Figure 3(i).
Since π is simple, regions A and B cannot both be empty, so, without loss of generality, suppose that A is non-empty and label the greatest entry in this region as point x.
Since π is simple, the rectangular hull of the leftmost entry, the least entry, and the point x cannot be an interval in π. Therefore, there must be a point either in B, or in that part of C lying below x. Take the rightmost such point, and label it y. If y is in region C, we immediately encounter the contradiction illustrated in Figure 3(ii): our choices of x and y, and the forbidden permutation 24153 causes the permutation to be sum decomposable. Therefore, y is placed in region B, and we have the picture depicted in Figure 3(iii). Since π is simple, there must now be a point in the region labelled D. However, labelling the greatest entry in this region yields another contradiction, as π must again be sum decomposable. The class Avp24153, 31524, ℓ¨¨¨21q is closed under group-theoretic inversion (because 24153´1 " 31524 and ℓ¨¨¨21´1 " ℓ¨¨¨21), so we may always assume that the latter option in Proposition 4.1 holds.
The rest of our proof adapts several ideas from Vatter [18]. Two rectangles in the plane are said to be dependent if their projections onto either the xor y-axis have nontrivial intersection, and otherwise they are said to be independent. A set of rectangles is called independent if its members are pairwise independent. Thus an independent set of rectangles may be viewed as a permutation, and it satisfies the Erdős-Szekeres Theorem (every permutation of length at least pa´1qpb´1q`1 contains either 12¨¨¨a or b¨¨¨21). We construct independent sets of rectangles in the proofs of both Propositions 4.3 and 4.5. In these settings, the rectangles are used to capture "bad" areas in the plot of a permutation, and our desired result is obtained by slicing the rectangles with horizontal and vertical lines in the sense of Proposition 3.3. The following result shows that we may slice a collection of rectangles with only a few lines, so long as we can bound its independence number. Proof. By Proposition 3.3 it suffices to show that there is a function gpℓq such that every permutation in Avp24153, 31524, ℓ¨¨¨21q can be covered by gpℓq increasing rectangles (i.e., rectangles which only cover increasing sets of points). We prove this statement by induction on ℓ. For the base case, we can take f p2q " 1. Now take a simple permutation π P Avp24153, 31524, ℓ¨¨¨21q for ℓ ě 3 and suppose that the claim holds for ℓ´1.
By Proposition 4.1, we may assume that the leftmost entry of π lies above its rightmost entry. Let π t be the permutation formed by all entries of π lying above its rightmost entry and π b the permutation formed by all entries of π lying below its leftmost entry (as shown in Figure 4). Thus every entry of π corresponds to an entry in π t , to an entry in π b , or to entries in both permutations. Moreover, both π t and π b avoid pℓ´1q¨¨¨21.
We would like to apply induction to find monotone rectangle coverings of both π t and π b but of course these permutations needn't be simple. Nevertheless, if π t is the inflation of the simple permutation σ t and π b of σ b then both σ t and σ b can be covered by gpℓ´1q increasing rectangles by induction. Now we stretch these increasing rectangles so that they cover the corresponding regions of π. By adapting the proof of Proposition 3.3, we may then extend this rectangle covering to a gridding of size at most 4gpℓ´1qˆ4gpℓ´1q. While this gridding needn't be monotone, inside each of its cells we see points which correspond either to inflations of increasing sequences of σ t or of σ b . Let L denote the grid lines of this gridding.
We now say that the axis-parallel rectangle R is bad if it is fully contained in a cell of the above gridding and the points it covers contain a decreasing interval in either π t or π b . Further let R denote the collection of all bad rectangles. We aim to show that there is a collection of f p2ℓpℓ1 qq lines which slice every bad rectangle, where f is the function defined in Theorem 4.2. This, together with Proposition 3.3 and the comments before it, will complete the proof because these lines together with L will give a gridding of π of bounded size. Theorem 4.2 will give us the desired lines if we can show that R has no independent set of size greater than 2ℓpℓ´1q`1. Suppose to the contrary that R does contain an independent set of this size. Thus at least ℓpℓ´1q`1 of these bad rectangles lie in one of π t or π b ; suppose first that these ℓpℓ´1q`1 bad rectangles lie in π t . Because π t avoids pℓ´1q¨¨¨21, the Erdős-Szekeres Theorem implies that at least ℓ`1 of its bad rectangles occur in increasing order (when read from left to right). Because π itself is simple, each such rectangle must be separated, and this separating point must lie in π t zπ b . Appealing once more to Erdős-Szekeres we see that two such separating points must themselves lie in increasing order, as shown in the centre of Figure 4. However, this is a contradiction to our assumption that π avoids 31524 (given by the solid points). As shown on the rightmost pane of this figure, if the ℓpℓ´1q`1 bad rectangles lie in π b we instead find a copy of 24153.
A submatrix of a matrix is obtained by deleting any collection of rows and columns from the matrix. Our next result shows that the simple permutations in Avp24153, 31524, ℓ¨¨¨21q can be gridded in a matrix which does not containˆ1 1 1˚˙a s a submatrix, i.e. in this matrix, there is no non-zero cell with both a non-zero cell below it in the same column, and a non-zero cell to its right in the same row. (The˚indicates an entry that can be either 0 or 1.) The following result is in some sense the technical underpinning of our entire argument. We advise the reader to note during the proof that if the hulls in H are assumed to be increasing, then the resulting gridding matrix M will be 0{1, not 0{˘1.  Proof. We define a gridding of π using the sides of the hulls in H. For a given side from a given hull, form a gridline by extending it to the edges of the permutation. By our hypothesis (H2), this line can slice at most k other hulls.
Whenever a hull is sliced in this way, a second gridline, perpendicular to the first, is induced so that all of the entries within the hull are contained in the bottom left and top right quadrants (for a hull containing increasing entries), or top left and bottom right quadrants (for a hull containing decreasing entries) defined by the two lines. This second gridline may itself slice through the interior of at most k further hulls in H, and each such slice will induce another gridline, and so on. We call this process the propagation of a line. See Figure 5 for an illustration. For a given propagation sequence, the propagation tree has gridlines for vertices, is rooted at the original gridline for the side, and has an edge between two gridlines if one induces the other in this propagation.
Before moving on, we note that it is clear that the propagation tree is connected, but less obvious that it is in fact a tree. This is not strictly required in our argument (we will only need to bound the number of vertices it contains), but if the tree were to contain a cycle it would have to correspond to a cyclic sequence of hulls, and this is impossible without contradicting hypothesis (H3).
In order to bound the size of a propagation tree, we first show that it has height at most 2m´1. In the propagation tree of a side from some hull H 0 P H, take a sequence H 1 , H 2 , . . . , H p of hulls from H corresponding to a longest path in the propagation tree, starting from the root. Thus, H 1 is sliced by the initial gridline formed from the side of H 0 , and H i is sliced by the gridline induced from H i´1 for i " 2, . . . , p.
We now define a word w " w 1 w 2¨¨¨wp from this sequence, based on the position of hull H i relative to hull H i´1 . For i " 1, 2, . . . , p, let Note that by the process of inducing perpendicular gridlines, successive letters in w must alternate between tu, du and tl, ru. Moreover, since no hull interacts with hulls both below it and to its right, w cannot contain a ur or ld factor. In other words, after the first instance of u or l, there are no more instances of r or d. This means that w consists of a (possibly empty) alternating sequence drdr¨¨¨or rdrd¨¨¨followed by a (possibly empty) alternating sequence ulul¨¨¨or lulu¨¨¨.
Any alternating sequence of the form drdr¨¨¨or rdrd¨¨¨can have at most m´1 letters, as each hull in H (other then H 0 ) can be sliced at most once in such a sequence. Similarly, any alternating sequence of the form ulul¨¨¨or lulu¨¨¨can have at most m´1 letters. Consequently, we have p ď 2m´2, and thus every propagation sequence has length at most 2m´1, as required.
Since each gridline in the propagation tree has at most k children, this means that the propagation tree for any given side has at most 1`k`k 2`¨¨¨`k2m´1 ă k 2m vertices, yielding a gridding of π with fewer than 4mk 2m gridlines, and we may take this number to be f pm, kq. The gridding matrix M is then naturally formed from the cells of this gridding of π: each empty cell corresponds to a 0 in M , each cell containing points in decreasing order corresponds to a´1 in M , and each cell containing points in increasing order corresponds to a 1 in M .
Finally, we verify that M satisfies the conditions in the proposition. The process of propagating gridlines ensures that each rectangular hull in H is divided into cells no two of which occupy the same row or column of the M -gridding. This means that there can be noˆ˘1˘1 1˚˙s ubmatrix of M with two cells originating from the same hull in H. Thus, the cells of a submatrix of the form˘1˘1 1˚˙m ust be made up from points in distinct hulls in H, but this is impossible since H contains no hulls which are dependent with hulls both below it and to its right.
We now apply this proposition to refine the gridding provided to us by Proposition 4.3. Proof. Let π be an arbitrary simple permutation in Avp24153, 31524, ℓ¨¨¨21q. As we observed in Footnote }, it suffices to show that there are constants a and b such that π P GridpM q for an aˆb matrix M satisfying the desired conditions. By Proposition 4.3, π is contained in GridpN q for some 0{1 matrix N , of size (say) tˆu. We say that a bad rectangle within any specified cell is an axis-parallel rectangle which contains two entries which are split both by points below and to the right. Since π does not contain 24153, no cell can contain more than one independent bad rectangle -see Figure 6. Therefore an independent set of bad rectangles can have size at most tu, so Theorem 4.2 shows that the bad rectangles can all be sliced by f ptuq lines.
In any cell of the original gridding, the additional f ptuq slices that have been added can at most slice these points into f ptuq`1 maximal unsliced pieces. In the entire permutation, therefore, these slices (together with the original gridlines from N ) divide the points into at most tupf ptuq`1q maximal unsliced pieces. Let H denote the rectangular hulls of these maximal unsliced pieces.
We now check that H satisfies the hypotheses of Proposition 4.4. Condition (H1) follows immediately by construction. Next, note that no two hulls of H from the same cell of the N -gridding can be dependent, since every cell is monotone, so we may take k " maxpt, uq to satisfy (H2). Finally, no hull can contain a bad rectangle (since all bad rectangles have been sliced), and so no hull can simultaneously be dependent with a hull from a cell below it, and a hull from a cell to its right, as required by (H3). Now, applying Proposition 4.4 (noting that all the rectangular hulls in H contain increasing entries), we have a 0{1 gridding matrix M π for π, of dimensions at most vˆw for some v, w, which does not containˆ1 1 1˚˙a s a submatrix. We are now done by our comments at the beginning of the proof.
Having proved Proposition 4.5, we merely need to put the pieces together to finish the proof of our main theorem. This proposition shows that there is a finite 0{1 matrix M with no submatrix of the formˆ1 1 1˚ṡ uch that the simple permutations of Avp24153, 31524, ℓ¨¨¨21q are contained in GridpM q. It follows that Avp24153, 31524, ℓ¨¨¨21q Ď xGridpM qy.
Moreover, M is a forest because if it were to contain a cycle, it would have to contain a submatrix of the formˆ1 1 1˚˙. Therefore the permutation class Avp24153, 31524, ℓ¨¨¨21q is wqo by Theorem 3.4. Theorem 4.6. For every ℓ, the permutation class Avp24153, 31524, ℓ¨¨¨21q is wqo. Therefore the class of permutation graphs omitting P 5 and K ℓ is also wqo.

PERMUTATION GRAPHS OMITTING
In this section, we establish the following: Proposition 5.1. The following three classes of graphs are not wqo: (1) the P 6 , K 6 -free permutation graphs, (2) the P 7 , K 5 -free permutation graphs, and (3) the P 8 , K 4 -free permutation graphs.
In order to prove these three classes are not wqo, it suffices to show that each class contains an infinite antichain. This is done by showing that the related permutation classes contain "generalised" grid classes, for which infinite antichains are already known. In general, we cannot immediately guarantee that the permutation antichain translates to a graph antichain, but we will show that this is in fact the case for the three we construct here.
We must first introduce generalised grid classes. Suppose that M is a tˆu matrix of permutation classes (we use calligraphic font for matrices containing permutation classes). An M-gridding of the permutation π of length n in this context is a pair of sequences 1 " c 1 ď¨¨¨ď c t`1 " n`1 (the column divisions) and 1 " r 1 ď¨¨¨ď r u`1 " n`1 (the row divisions) such that for all 1 ď k ď t and 1 ď ℓ ď u, the entries of π from indices c k up to but not including c k`1 , which have values from r ℓ up to but not including r ℓ`1 are either empty or order isomorphic to an element of M k,ℓ . The grid class of M, written GridpMq, consists of all permutations which possess an M-gridding. The notion of monotone griddability can be analogously defined, but we do not require this.
Here, our generalised grid classes are formed from gridding matrices which contain the monotone class Avp21q, and a non-monotone permutation class denoted À 21. This is formed by taking all finite subpermutations of the infinite permutation 21436587¨¨¨. In terms of minimal forbidden elements, we have À 21 " Avp321, 231, 312q.
Our proof of Proposition 5.1 requires some further theory to ensure we can convert the permutation antichains we construct into graph antichains. The primary issue is that a permutation graph G can have several different corresponding permutations. With this in mind, let ΠpGq " tpermutations π : G π -Gu denote the set of permutations each of which corresponds to the permutation graph G.
Denote by π´1 the (group-theoretic) inverse of π, by π rc the reverse-complement (formed by reversing the order of the entries of π, then replacing each entry i by |π|´i`1), and by pπ´1q rc the inversereverse-complement, formed by composing the two previous operations (in either order, as the two operations commute). It is then easy to see that if π P ΠpGq, then ΠpGq must also contain all of π´1, π rc and pπ´1q rc . However, it is possible that ΠpGq may contain other permutations, and this depends on the graph-theoretic analogue of the substitution decomposition, which is called the modular decomposition.
We also need to introduce the graph analogues of intervals and simplicity, which have different names in that context. A module M in a graph G is a set of vertices such that for every u, v P M and w P V pGqzM , u is adjacent to w if and only if v is adjacent to w. A graph G is said to be prime if it has no nontrivial modules, that is, any module M of G satisfies |M | " 0, 1, or |V pGq|.
The following result, arising as a consequence of Gallai's work on transitive orientations, gives us some control over ΠpGq: Proposition 5.2 (Gallai [9]). If G is a prime permutation graph, then, up to the symmetries inverse, reversecomplement, and inverse-reverse-complement, ΠpGq contains a unique permutation.
We now extend Proposition 5.2 to suit our purposes. Consider a simple permutation σ of length m ě 4, and form the permutation π by inflating two˚˚of the entries of σ each by the permutation 21.˚I t is, in fact, possible to inflate more than two entries and establish the same result, but we do not require this here. Note that G 21 " K 2 , and ΠpK 2 q " t21u. In the correspondence between graphs and permutations, modules map to intervals and vice versa, so prime graphs correspond to simple permutations, and there is an analogous result to Proposition 3.5 for (permutation) graphs.
Thus, for any permutation ρ P ΠpG π q, it follows that ρ must be constructed by inflating two entries of some simple permutation τ by the permutation 21. Moreover, τ must be one of σ, σ´1, σ rc or pσ´1q rc , and the entries of τ which are inflated are determined by which entries of σ were inflated, and which of the four symmetries of σ is equal to τ . In other words, we still have ΠpG π q " tπ, π´1, π rc , pπ´1q rc u.
We require one further easy observation: Lemma 5.3. If G and H are permutation graphs such that H ď G, then for any π P ΠpGq there exists σ P ΠpHq such that σ ď π.
Proof. Given any π P ΠpGq, let σ denote the subpermutation of π formed from the entries of π which correspond to the vertices of an embedding of H as an induced subgraph of G. Clearly G σ -H, so σ P ΠpHq and σ ď π, as required.
We are now in a position to prove Proposition 5.1. Since the techniques are broadly similar for all three cases, we will give the details for case (2), and only outline the key steps for the other two cases.
Proof of Proposition 5.1 (2). First, the graph P 7 corresponds to two permutations, namely 3152746 and 2416375 respectively. Thus the class of P 7 , K 5 -free permutation graphs corresponds to the permutation class Avp3152746, 2416375, 54321q. This permutation class contains the grid class because this grid class avoids the permutations 241635 (contained in both 3152746 and 2416375), and 54321.
We now follow the recipe given by Brignall [7] to construct an infinite antichain which lies in Grid p À 21 À 21 q. Call the resulting antichain A, the first three elements and general term of which are illustrated in Figure 7. This antichain is related to the "parallel" antichain in Murphy's thesis [15]. Now set G A " tG π : π P Au, and note that G A is contained in the class of permutation graphs omitting P 7 and K 5 by Lemma 5.3. If G A is an antichain of graphs, then we are done, so suppose for a contradiction that there exists G, H P G A with H ď G. Take the permutation π P A for which G π " G. Applying Lemma 5.3, there exists σ P ΠpHq such that σ ď π. We cannot have σ P A since A is an antichain, so σ must be some other permutation with the same graph. Choose τ P AX ΠpHq.
It is easy to check that the only non-trivial intervals in any permutation in A are the two pairs of points in Figure 7 which are circled. Thus, τ is formed by inflating two entries of a simple permutation each by a copy of 21. By the comments after Proposition 5.2, we conclude that σ is equal to one of τ´1, τ rc or pτ´1q rc .
Note that every permutation in A is closed under reverse-complement, so τ " τ rc and thus σ ‰ τ rc . If σ " τ´1, by inspection every element of A contains a copy of 24531, which means that σ contains p24531q´1 " 51423. Since σ ď π, it follows that π must also contain a copy of 51423. However this is impossible, because 51423 is not in Grid p À 21 À 21 q.
Thus, we must have σ " pτ´1q rc , in which case σ must contain a copy of pp24531q rc q´1 " 34251. However, this permutation is also not in Grid p À 21 À 21 q so cannot be contained in π. Thus σ R π, and from this final contradiction we conclude that H ę G, so G A is an infinite antichain of permutation graphs, as required.
By [7], this grid class contains an infinite antichain whose first three elements are illustrated in Figure 8. The permutations in this antichain have exactly two proper intervals, indicated by the circled pairs of points in each case, and this means that for any permutation π in this antichain, ΠpG π q " tπ, π´1, π rc , pπ´1q rc u.
Following the proof of Proposition 5.1 (2), it suffices to show that for any permutations σ and π in the antichain, none of σ, σ´1, σ rc , or pσ´1q rc is contained in π. This is done by identifying permutations which are not contained in any antichain element π, but which are contained in one of the symmetries σ´1, σ rc , or pσ´1q rc . We omit the details.
For (3), The P 8 , K 4 -free permutation graphs correspond to Avp4321, 24163857, 31527486q, and this permutation class contains the (monotone) grid class GridˆA vp21q Avp21q Avp21q Avp21q˙. In this case, we appeal to Murphy and Vatter [16] for an infinite antichain, and the same technique used to prove cases (1) and (2) can be applied.

ENUMERATION
One significant difference between studies of the induced subgraph order and the subpermutation order is the research interests of the different camps of investigators. In the latter area, a great proportion of the work is enumerative in nature. Therefore, having established the structure of permutation graphs avoiding P 5 and cliques, we briefly study the enumeration of the corresponding permutation classes in this section. Our goal is to show that these classes are strongly rational, i.e., that they and every one of their subclasses have rational generating functions. Here we refer to ÿ πPC x |π| as the generating function of the class C, where |π| denotes the length of the permutation π. Note that by a simple counting argument (made explicit in Albert, Atkinson, and Vatter [3]), strongly rational permutation classes must be wqo.
Proposition 3.5 allows us to associate with any permutation π a unique substitution decomposition tree. This tree is recursively defined by decomposing each node of the tree as • σrα 1 , . . . , α m s where σ is a nonmonotone simple permutation, See Figure 9 for an example. In particular, note that by our indecomposability assumptions, sum nodes (resp., skew sum nodes) cannot occur twice in a row when reading up a branch of the tree. The substitution depth of π is then the height of its substitution decomposition tree, so for example, the substitution depth of the permutation from Figure 9 is 3, while the substitution depth of any simple (or monotone) permutation is 1. As we show in our next result, substitution depth is bounded for the classes we are interested in. This result is a special case of Vatter [18,Proposition 4.2], but we include a short proof for completeness.
Proof. We prove the result using induction on ℓ. Only increasing permutations avoid 21 and they have substitution depth 1, so the result holds for ℓ " 2. Suppose to the contrary that the result holds for ℓ ě 2 but that there is a permutation π P Avppℓ`1q¨¨¨21q of substitution depth 2ℓ (we may always assume that the depth is precisely this value, because of downward closure). If π is a sum decomposable permutation, express it as π " π 1 '¨¨¨' π m and note that at least one π i must have substitution depth 2ℓ´1.
Thus we may now assume that there is a sum indecomposable permutation π P Avppℓ`1q¨¨¨21q of substitution depth 2ℓ´1. Write π " σrα 1 , . . . , α m s where σ is either a simple permutation or decreasing of length at least 2 (to cover the case where π is skew decomposable). At least one of the α i must have substitution depth 2ℓ´2, and contains ℓ¨¨¨21 by induction. However, there is at least one entry of σ which forms an inversion with σpiq, and thus π itself must contain pℓ`1q¨¨¨21, as desired.
It was shown in Albert, Atkinson, Bouvel, Ruškuc and Vatter [2, Theorem 3.2] that if M is a forest, then GridpM q is a geometric grid class. In the same paper, two strong properties of geometric grid classes were established: they are defined by finitely many minimal forbidden permutations and they are strongly rational. We refer the reader to that paper for a comprehensive introduction to geometric grid classes. Our enumerative result follows from the following theorem proved in a subsequent paper. Applying induction on the height of substitution decomposition trees (which are bounded in all of our classes by Proposition 6.1), we immediately obtain the following result. Theorem 6.3. For every ℓ, the class Avp24153, 31524, ℓ¨¨¨21q is strongly rational.

CONCLUDING REMARKS
As shown in Figure 1, there are only three cases remaining: permutation graphs avoiding tP 6 , K 5 u, tP 6 , K 4 u and tP 7 , K 4 u. Due to the absence of an "obvious" infinite antichain in these cases, we conjecture that they are all wqo. However, all three classes contain (for example) the generalised grid class Grid p À 21 Avp21q q, so these classes all contain simple permutations which are not monotone griddable. Thus our approach would have to be significantly changed to approach this conjecture.
A natural first step might be to show that the corresponding permutation classes are D-griddable (in the sense of Section 5) for a "nice" class D. As any proof along these lines would need to use some sort of slicing argument as well (such we used to prove Proposition 4.5), D would have to be nice enough to allow for such arguments. Vatter [18,Lemma 5.3] has shown that such slicing arguments can be made to work when D has only finitely many simple permutations and finite substitution depth (in the sense of Section 6). As we already have finite substitution depth by Proposition 6.1, it would suffice to show that we could take D to contain only finitely many simple permutations.
Again, though, this would only be an encouraging sign and not a proof, because one would then have to develop more sophisticated tools to prove wqo for such classes. Specifically, it is unlikely that the existing machinery of Brignall [7] would suffice.