Unifying duality theorems for width parameters in graphs and matroidsII. General duality

Unifying duality theorems for width parameters in graphs and matroids
II. General duality

Reinhard Diestel    Sang-il Oum The second-named author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (2011-0011653).
Abstract

We prove a general duality theorem for tangle-like dense objects in combinatorial structures such as graphs and matroids. This paper continues, and assumes familiarity with, the theory developed in [6].

1 Introduction

This is the second of two papers on the duality between certain ‘dense objects’ in a combinatorial structure such as a graph or a matroid, and tree-like structures to which any graph or matroid that does not contain such a ‘dense object’ must conform. Its ‘tree-likeness’ is then measured by a so-called width parameter, which is the smaller the more the graph or matroid conforms to such a tree-shape. The ‘dense objects’, which traditionally come in various guises, are in our framework cast uniformly in a way akin to tangles: as orientations of certain separation systems of the graphs.

In Part I of this paper [6] we proved a duality theorem which unifies and extends the duality theorems for the classical width parameters of graphs and matroids, such as branch-width, tree-width and path-width. Our aim, however, had not been to find such a theorem. What we were looking for was a duality theorem for more general width parameters still, one that would also cover more recently studied tangle-like objects such as ‘blocks’ [5, 4] and ‘profiles’ [7, 2, 3].

Amusingly, our theorem from [6] covers neither blocks nor profiles, although it does cover all those other parameters, and blocks are similar to tangles while profiles are sandwiched between tangles and brambles (and generalize blocks). We shall see in this paper why our attempt had to fail: we prove that both blocks and profiles need more general obstructions to witness their non-existence than the structure trees used in [6].

As our main positive result, we shall prove a duality theorem that does cover such general tangle-like objects as blocks and profiles (and countless others). Since the obstructions it identifies for their non-existence are more general than trees, it will not imply our results from [6] nor follow from them.

We shall use the same terminology as in [6]; see Section 2 for what exactly we shall assume the reader is familiar with. In Section 3 we describe the general tangle-like structures we shall cover, while in Sections 45 we describe the more general tree-like structures to witness their non-existence. Our General Duality Theorem will be proved in Section 6. In Section 7 we apply it to blocks and profiles (which are also defined formally there), and show that for these structures our duality theorem is best possible: there are graphs that have neither a block or profile nor admit a tree-like decomposition as in the duality theorem of [6], thus showing that the more general tree-like structures employed by our main result are best-possible not only for a general duality theorem but also for just blocks and profiles.

2 Background needed for this paper

We assume that the reader is familiar with the terminology set up in Part I [6, Section 2], and with the proof of its Weak Duality Theorem [6, Section 3]. Let us restate this theorem:

Theorem 2.1 (Weak Duality Theorem).

Let be a separation system of a set , and let contain every separation of the form . Let be a set of stars in . Then exactly one of the following holds:

  1. There exists an -tree over rooted in .

  2. There exists an -avoiding orientation of extending .

The conclusion of the Strong Duality Theorem [6] differs from this in one crucial detail: it asks that the orientations in (ii) be consistent, that they never orient two separations away from each other.111…which indeed makes little sense if we think of these orientations as pointing to some dense object. This comes at the price of having to restrict and  in the premise, but we shall not need the technicalities of this restriction in this paper.

As pointed out earlier, our motivation for starting this line of research was to find a duality theorem, and associated width parameter, for two notions of ‘dense objects’ that have recently received some attention, so-called blocks and profiles (defined at the start of Section 7).

As it turns out, these cannot be captured in the framework developed in [6], as (consistent) orientations of separation systems avoiding a certain collection  of stars of separations. Our General Duality Theorem will therefore relax this requirement by allowing to contain arbitrary ‘forbidden’ sets of separations. However, we shall see in Section 3 that it will suffice to consider so-called ‘weak stars’ as elements of .

In Sections 45 we deal with the other side of the duality, the tree-structure. It turns out that this, too, has to be relaxed for any duality theorem for blocks or profiles. Of course, we would have to allow -trees over weak stars rather than just over stars as in [6], but even this is not enough: we need to relax the trees to certain graphs with few cycles, which we shall call -graphs.

3 From stars to weak stars

Recall that a set of separations of a set is a star if these separations are nested and point towards each other, that is, if for all distinct . In both the Weak and the Strong Duality Theorem proved in [6], the sets of separations that were forbidden in the orientations of a separation system defining a particular ‘dense object’, those in , were stars.

In the General Duality Theorem we shall prove here,  can be an arbitrary collection of sets of separations. It will be good, however, to be able to restrict this arbitrariness if desired: the smaller we can make , the easier will it be to show that an orientation is -avoiding. In this section we show that can always be restricted to the ‘weak stars’ it contains.

Figure 1: The separations in (i) form a weak star; those in (ii) and (iii) do not.

A set of separations is a weak star if it is a consistent -antichain (Fig. 1). Clearly, antisymmetric222A set of separations is antisymmetric if whenever . stars of proper separations are weak stars, and a weak star is a star if and only if it is nested. Given a set of separations, let denote the set of its -maximal elements. Then, for any collection of sets of separations, all the elements of

are weak stars. Note that, formally,  need not be a subset of . But in all our applications it will be, and if it is, it is just the subset of consisting of all its weak stars.

Given two sets of separations of , let us define

This is a reflexive and transitive relation on the sets of separations of , and on the weak stars it is also antisymmetric. Given any collection of sets of separations, we let

Lemma 3.1.

The following statements are equivalent for every consistent orientation of a finite separation system of  and .

  1. avoids .

  2. avoids .

  3. avoids .

Proof.

(i)(ii)(iii): Consider any set . If then also , because is a consistent orientation of , and every separation in  lies below some separation in . Hence if avoids  it also avoids , and thus also .

(iii)(ii): If then , since is a consistent orientation of . Since for every there exists an in , this proves the assertion.

(ii)(i): If has a subset in , then is consistent, so lies in . Hence if avoids , it must avoid . ∎

4 Obstructions to consistency

As pointed out earlier, the key advance of our Strong Duality Theorem over the weak one is that the orientations of the given separation system whose existence it claims will be consistent. This comes at a price: to obtain consistent orientations in the proof we had to impose a condition on and , that should be ‘-separable’.

The following simple example shows that imposing some condition was indeed necessary: even if consists only of stars, it can happen that there is neither a consistent -avoiding orientation of  nor an -tree over  that extends some given consistent .

Example 4.1.

Let consist of two pairs of crossing separations and their inverses: and , such that for all choices of . Let , and let contain the stars and (Fig. 2).

There exists a unique -avoiding orientation of extending , which contains to avoid in the presence of , as well as to avoid in the presence of . But these two separations, and , face away from each other, so this orientation of is inconsistent.

However, there is no -tree over  rooted in . Indeed, the only star in containing is , the only star in containing is , and there are no further stars in . Figure 2 shows the forest that arises instead of an -tree on the right.

Figure 2: A separation system that has no consistent orientation extending , but also no -tree over  rooted in . Its ‘-forest’ is shown on the right.

However, we can repair the ‘-forest’  by generalizing the notion of an -tree, as follows. The idea of an -tree is that it should witness the non-existence of certain orientations of . The non-existence of a consistent orientation of  should be easier to witness, because it is a weaker property. And indeed, we can endow our -trees with an additional feature to witness violations of consistency: nodes of degree 2 whose incoming edges map to separations that point away from each other.333This is the opposite of requiring the incoming edges at a node to map to a star of separations, which would have these point towards each other. In particular, if we allow such new nodes of degree 2, then will no longer preserve the natural orientation of . As the reader may check, a tree whose oriented edges map to separations in such a way that (incoming) stars at nodes either map to stars in and or are 2-stars of this new type will still witness the non-existence of a consistent -orientation avoiding  (where, as before, leaf separations must be in , which in turn must be included in any -orientation considered): any consistent -avoiding orientation of will induce, via , an orientation of in which no interior node of  is a sink; so there has to be a sink at a leaf, implying if is rooted in .

Figure 3: Constructing a generalized -tree

Figure 3 shows such a ‘generalized -tree’. We shall build on this idea later when we define ‘-graphs’ over arbitrary sets , the objects dual to consistent -avoiding -orientations in our General Duality Theorem.

Alternatively, one might suspect that the failure of duality in Example 4.1 stems from our unhelpful choice of . Our next example of a separation system has no consistent orientation whatsoever avoiding a certain (again consisting of stars), none extending even . But as there are no trees without leaves, it cannot have an -tree rooted in  either, not even one generalized as above.

Example 4.2.

Let consist of five inverse pairs of separations, arranged cyclically so that each crosses its predecessor and its successor but is nested with the other two pairs of separations, as shown in Fig. 4. Let . Let consist of all the stars in , each consisting of two separations pointing towards each other. Then in any consistent -avoiding orientation of  every two nested separations will be comparable: they will not point towards each other because avoids , and they will not point away from each other because is consistent. We shall prove that has no such orientation . Since , it cannot have an -tree either, not even one generalized as above.

For each , let be the separation from the pair  that lies in . Let be the graph on in which two form an edge whenever they are nested. This is a 5-cycle; pick one of its two orientations. Since nested separations in are comparable, we have for each oriented edge in either , in which case we colour the edge green, or , in which case we colour it red. Since 5 is odd,  has two equally coloured edges, so contains three adjacent separations . But there are no three pairwise nested separations in , a contradiction.

Figure 4, right, shows an attempted ‘generalized -tree’ – in fact, a 10-cycle – pieced together by (incoming) 2-stars mapping to  (solid nodes) or, alternately, witnessing the inconsistency of two separations (hollow vertices).

Figure 4: Cyclically arranged separations not defining a generalized -tree

Note that our proof of the non-existence of a consistent -avoiding orientation in Example 4.2 was quite ad-hoc and perhaps not obvious. As an application of our General Duality Theorem, we shall later see a more canonical proof, illustrated by Figure 8.

5 From -trees to -graphs

The essence of our duality theorems is that if a given separation system has no -avoiding orientation (possibly consistent) that extends a given set , then this is witnessed by a particularly simple subset of : a nested subsystem that already admits no -avoiding orientation extending . Since nested separation systems define tree-like decompositions of the structures they separate, it is convenient to describe them as -trees.

As we saw in Section 4, however, it can happen for certain choices of and  that admits no consistent -avoiding orientation but every nested subsystem of does. In such cases, it may still be possible to find a subsystem of which, though perhaps not nested, is still considerably simpler than and also admits no consistent -avoiding orientation, and which can thus be used as a witness to the fact that admits no such orientation.

It is our aim in this section to present a structure type for separation subsystems, slightly more general than nested systems, that can always achieve this. These structures are formalized as -graphs, a generalization of -trees just weak enough to describe such systems when they are not nested. Both -trees and the ‘generalized -trees’ considered in Section 4 will be examples of -graphs.

In order to illuminate the idea behind their definition, let us recall the standard proof of why any -tree over rooted in  is an obstruction to -avoiding orientations . Via , the orientation of  orients the edges of . Since the stars at nodes of map to stars in , which are never subsets of , the edges of at a given node are never all oriented towards it. Similarly, since is rooted at , no edge of is oriented towards a leaf. Hence has no sink in this orientation, which cannot happen.

In this proof we did not use that is a tree except at the end, when we needed that no orientation of can leave it without a sink. And neither did we use that the separations in the image of are nested. An -graph, in the same spirit, will be a graph with a map from its edge orientations to  such that any consistent -avoiding orientation of will orient the edges of in a way that contradicts its structure.

The following types of graph will be used as -graphs. Consider finite connected bipartite undirected graphs with at least one edge and bipartition . For every vertex let its set of incident edges be partitioned into two non-empty classes, as . Call every vertex of degree 1 a leaf, and assume that all leaves lie in . Let be the class of all such graphs.

Let be a separation system of a set , and let and . An -graph over rooted in  is a pair such that is a graph in , with bipartition say, and satisfies the following:

  1. commutes with inversions of edges in and of separations in , i.e., implies ;

  2. the incident edge of any leaf satisfies ;

  3. for every node that is not a leaf, with incident edges say, ;

  4. whenever and , .

Note that (i)–(iii) are copied from the definition of an -tree over  rooted in . In particular, (iii) makes map oriented stars at nodes to ‘forbidden’ sets of separations in . Similarly, (iv) makes map oriented stars at vertices to pairs of separations that violate consistency (Fig. 5). We shall refer to (ii) by saying that is rooted in , to (iii) by saying that is over , and to (iv) by saying that the edges of at vertices witness inconsistencies.

Figure 5: At , the map witnesses the inconsistency of with , as .

Thus, any -tree over rooted in  becomes an -graph over rooted in  on subdividing every edge by a vertex  and letting . And so do the ‘generalized -trees’ considered in Section 4 if we subdivide the original edges of their underlying -tree.444Not their new edges witnessing violations of consistency, which already come in pairs sharing a vertex of degree 2 and satisfy (iv).

Note also that condition (iv) is invariant under swapping the names of and , since implies by (i). The only purpose of partitioning into these two sets is to be able to define ‘traversing’ below; it does not matter which of the two partition sets is  and which is .

Finally, we remark that condition (ii) could be subsumed under (iii) by putting the inverses of separations in in as singleton sets and applying (iii) also to leaf nodes : since any orientation of must include every separation from or its inverse, forbidding the inverses of separations in  amounts to including . However, it will be convenient in the proof of our duality theorem to treat the two separately.

The -graphs we shall in fact need will have some further properties that make them more like trees. Let us say that a path or cycle in a graph as above traverses if it has an edge in and another in . We shall call , and any -graph based on it, cusped555The word ‘cusped’ is intended to convey a notion of ‘nearly as spiky as a tree’: by (ii) below, any cycle in a cusped graph must have a ‘cusp’ at a vertex , entering and leaving it through the same partition class of . if

  1. every edge such that is a leaf is the only edge in its bipartition class of ;

  2. no cycle in traverses all the vertices of that it contains.

Note that non-trivial trees become cusped graphs if we subdivide every edge.

We shall prove that cusped -graphs over that are rooted in  are obstructions to consistent -avoiding orientations extending . The following property of cusped graphs is at the heart of that proof:

Lemma 5.1.

Let be a cusped graph, with bipartition classes and for all . For every orientation of its edges,  either has a node with all incident edges oriented towards , or it has a vertex such that both and  contain an edge oriented towards .

Proof.

In a given orientation of , let be a maximal forward-oriented path that starts at a node in  and traverses every it contains unless it ends there.

Suppose first that ends at a vertex . Then has an edge in only one of the two partition classes of . If all the edges in the other partition class are oriented towards , then has the desired property, because that other partition class is also non-empty (by definition of ). If not, the other partition class contains an edge oriented towards  and with . Adding this edge to the final segment of  we obtain a cycle in that traverses all its vertices in , a contradiction.

Suppose now that ends at a node . If all the edges of at are oriented towards , then is as desired. If not, there is an edge oriented away from . By the maximality of we cannot append this edge to , so traverses . Since the cycle obtained by adding the edge to the final segment of does not traverse all its vertices in , the first edge of lies in the same partition class as . Then the edge preceding on  and the edge are both oriented towards  and lie in different partition classes of , as desired. ∎

We remark that the converse of Lemma 5.1 can fail: the graph shown in Figure 6, in which the two hollow vertices are in and their incident edges are partitioned into ‘left’ and ‘right’, satisfies the conclusion of the lemma but contains a cycle that traverses all its vertices in .

Figure 6: A counterexample to the converse of Lemma 5.1

Although we shall prove that any cusped -graph can be used as a witness to the non-existence of the corresponding orientations of , we shall also prove that there are always witnesses among these that can be constructed in a particularly simple way: recursively from subdivided -trees by amalgamations in a single vertex. Let us call a graph in constructible if either

  1. is obtained from a -star on  with by subdividing every edge once and putting the subdividing vertices in ; or

  2. is obtained from the disjoint union of two constructible graphs as follows. Let and with the familiar notation. Let be a non-empty set of leaves of  such that, for every , its incident edge  is the unique edge in its partition class of , and let be an analogous set of leaves in . Let be obtained from by identifying all the neighbours in  of nodes in  with all the neighbours in  of nodes in  into a new vertex  (Fig. 7). Let be the set of vertices of that are in or equal to , and let be the set of all other vertices of  (those in ). Let consist of the edges at  that come from , and let consist of the edges coming from .

Figure 7: The recursion for constructible graphs
Lemma 5.2.

Constructible graphs are cusped.

Proof.

We apply induction following their recursive definition. Since subdivided trees are cusped, the induction starts. Now let be obtained from two cusped graphs as in (P2). Since is a cutvertex of  dividing and into different blocks, no cycle of through traverses . Hence inherits property (ii) from the definition of ‘cusped’ from the graphs .

To check property (i), consider an edge of such that is a leaf of . Assume that . Then is a leaf also in . Then the partition class of its incident edge in , say , contains only . But by construction of , the other partition class of edges at  in  also contains only one edge , with a leaf of . Since is connected, this means that is just the 2-path . Hence , so is the only edge in its partition class also in . ∎

We remark that while many cusped graphs are constructible [8], not all are. For example, a vertex in a constructible graph will never separate the other ends of two of its incident edges from the same partition class of , but this can happen in an arbitrary cusped graph such as a tree.

We can now construct -graphs in the same way. Let be an -graph over , with and partitions of the sets as earlier. We say that is constructible (over ) if either

  1. is obtained from an -tree over  with a -star on  () by subdividing every edge once, putting the subdividing vertices in , and letting whenever subdivides the edge ; or

  2. is obtained as in (P2) from the disjoint union of two graphs in -graphs and constructible over , in such a way that there exists a separation such that

    • for every with incident edge and every with incident edge ;

    • neither nor is a leaf separation666These are separations with a leaf [6]. in ;

(see Figure 7). Any -graph arising as in (S2) will be said to have been obtained from and by amalgamating with .

Thus in (S2), is obtained from and  by identifying, for some , all the leaves of with leaf separation with all the leaves of with leaf separation  and contracting all the edges at the identified node into one new amalgamation vertex .

Checking that as obtained in (S2) is again an -graph is straightforward: conditions (i)–(iii) from the definition of -graphs carry over from and , while (iv) holds because whenever and there are and such that

where the inequalities hold by (iv) for and . (In Figure 7, for example, we have in , because in  and in .)

6 The General Duality Theorem

We can now state and prove the most general version of our duality theorem. It looks for consistent orientations and allows in  arbitrary subsets (equivalently by Lemma 3.1, weak stars) rather than just stars. On the dual side it offers only -graphs rather than -trees as witnesses when such an orientation does not exist, but we can choose whether we want to use constructible or arbitrary cusped -graphs.

Theorem 6.1.

Let be a finite separation system. Let and . Then the following assertions are equivalent:

  1. There exists a consistent -avoiding orientation of extending .

  2. There is no cusped -graph over rooted in .

  3. There is no -graph over and rooted in  that is constructible over .

By Lemma 3.1, we may replace in (i) the set with the set of weak stars or the set of minimal weak stars in  to obtain an equivalent assertion; we may then leave (ii) and (iii) unchanged or change there as well, as we wish.

By the same argument, it will not be possible to narrow the class of -graphs allowed in (ii) and (iii) to any inequivalent subclass just by restricting in this way. For example, since Theorem 6.1 fails when we replace ‘-graph’ with ‘-tree’ (Example 4.2), it will still fail with ‘-tree’ when we restrict to or .

Proof of Theorem 6.1.

(i)(ii) Let be a consistent -avoiding orientation of , and suppose there is an -graph over  rooted in . Let be given with partitions and at vertices , as in the definiton of cusped graphs. Now consider the orientation of that induces via : orient an edge from to  if , and from to  if (and hence ).

The fact that extends  while is rooted in , the fact that avoids  while is over , and the fact that is consistent while the edges of at vertices can witness inconsistency, then imply the following:

  • at every node at least one incident edge is oriented away from ;

  • at every vertex either all edges in or all edges in are oriented away from .

This contradicts Lemma 5.1.

(ii)(iii) follows from Lemma 5.2.

(iii)(i) Suppose first that contains separations witnessing inconsistency, with say. Let be a path , and put and . Let and , as well as and . Then is an -graph as in (iii), completing the proof.

We now assume that contains no such . Then is a consistent partial orientation777A partial orientation of is an orientation of a symmetric subset of  [6]. of . We apply induction on .

If , then itself is an orientation of  extending . If (i) fails, then has a subset . Let be the vertex set of an -star with centre  and leaves . Subdivide every edge by a new vertex , and put these in . Let and , for . Then is an -graph as in (iii), completing the proof.

Thus we may assume that has a separation such that neither nor is in . Let and . Since any orientation of  extending or  also extends , we may assume that there is no such orientation that is both consistent and avoids .

By the induction hypothesis, or trivially if or is inconsistent, there are constructible -graphs and over , rooted in and , respectively. Unless one of these is in fact rooted in , contradicting (iii),  has at least one leaf , with neighbour  say, such that , and has at least one leaf , with neighbour  say, such that .

Let be the set of all these leaves of , and let be the set of all these leaves  of . It is easily verified that the -graph obtained from and by amalgamating with is as in (iii). ∎

Figure 8: An -graph corresponding to a cyclic arrangement of separations

Figure 8 shows an -graph witnessing the non-existence of a consistent orientation avoiding  in Example 4.2 (Fig. 4). It was found as the proof of Theorem 6.1 would suggest: not knowing what to do with we tentatively considered and , found -graphs over rooted in and , respectively (in fact, ‘generalized -paths’ as in Section 4; Fig. 9), and pieced these together to form the rootless -graph shown in Figure 8.

Figure 9: The -paths rooted in and

Notice that while it is easy to use this -graph, once found, to prove that has no consistent -avoiding orientation, it was not so easy to find it. In general, finding either a consistent -avoiding orientation of  or a cusped -graph over  is a hard problem: as Bowler [1] observed, the problem to decide whether a set of separations in a graph has a consistent orientation that avoids a given is NP-hard even when is fixed as  and the sets in consist of only three separations (and and are input).

7 Applications: blocks and profiles

One of the motivations for this paper was to find a duality theorem for two notions of ‘dense objects’ that received some attention recently, blocks and profiles.

-block in a graph , where is any positive integer, is a maximal set of at least  vertices such that no two vertices can be separated in  by fewer than  vertices other than and . A -profile of is a consistent orientation of the set of separations of order in such that whenever and then . Every -block induces a -profile (see below on how); tangles of order  are other examples of -profiles. Every -profile, in turn, is a haven of order  and thus gives rise to a bramble of order at least . Thus, profiles lie between blocks and brambles, but also between tangles and havens. See [5, 4, 7, 2, 3] for more on blocks and profiles.

Before we look at blocks and profiles separately, let us note that any consistent orientation of  extends

since would violate consistency by .

Let us look at blocks first. Consider a -block in a graph . For every separation we have or , but not both (since ). Thus,

is a consistent orientation of . Clearly,  has no subset in

(1)

so for it is an -avoiding consistent orientation of .

Conversely, every -avoiding orientation of  is clearly consistent, and

is a -block: no separation in separates any of its vertices, since that separation or its inverse lies in  and hence has a side containing ; and since because avoids , we have by (1).

We can thus obtain orientations of from -blocks, and vice versa. These operations are inverse to each other:

Lemma 7.1.

if and only if .

Proof.

Assume first that . Then consists of all the separations such that . So the intersection of all those  contains  too, and this intersection is . Thus, .

Conversely, since is maximal as an -inseparable set of vertices, any vertex is separated from some vertex of  by a separation in , and hence also by some separation . Since by definition of , this means that . So lies outside , and hence also outside the intersection of all with . Thus, .

Assume now that . For every , its ‘large side’ trivially contains the intersection of all such ‘large sides’ of separations in . So and hence , proving .

Conversely, if , then . Since is a -block and hence , this means that . Since would imply , by definition of , we thus have and hence , as desired. ∎

Thus, -blocks ‘are’ precisely the consistent -avoiding orientations of . When we now treat -blocks in our duality framework, we shall use the term ‘-block’ also to refer to these orientations.

Rephrasing -blocks as orientations of  throws up an interesting connection between blocks and brambles or havens that had not been noticed before. As is closed under taking subsets, it contains the set defined in Section 3:

Hence by Proposition 3.1 and Lemma 7.1, the -blocks of are precisely the consistent orientations of that contain no weak star from . If we delete the word ‘weak’ in this sentence, we obtain the consistent orientations of  that avoid

which are precisely the dual objects to tree-decompositions of width .888As shown in [6], has a haven of order at least , or equivalently a bramble of order at least , if and only if has a consistent -avoiding orientation.

Our General Duality Theorem thus specializes to blocks as follows:

Theorem 7.2.

For every finite graph and the following statements are equivalent:

  1. contains a -block.

  2. has a -avoiding orientation (which is consistent and extends ).

  3. has a consistent -avoiding orientation (which extends ).

  4. There is no -graph over  rooted in .

  5. There is no -graph over  rooted in .∎

We remark that, by Theorem 6.1, the -graphs in (iv) and (v) can be chosen to be constructible over  or , respectively.

To wind up our treatment of blocks, let us mention a quantitative999rather than structural, as our Theorem 7.2 duality theorem for blocks proved in [4]. There, the block-width of  is defined as the least integer  such that can be divided recursively into parts of size at most  by separations of  of order at most . (See [4, Section 7] for details.) It is not hard to show that this can be done if and only if has no -block for . The block number

thus comes with the following quantitative duality [4]:

Proposition 7.3.

Every finite graph satisfies .∎

Let us now turn to profiles. A -profile of is a consistent orientation of satisfying

This indirect definition is convenient, because it is not clear whether the separation is in , i.e., has order : if it does, it must be in , but if it does not, then there is no requirement. So the -profiles of are precisely the consistent orientations of  that avoid

Let and be obtained from as defined in Section 3. While crossing sets will be weak stars, some nested ones are not; in particular,  may be inconsistent. However, if with , then , so .

Our General Duality Theorem specializes to profiles as follows:

Theorem 7.4.

For every finite graph and the following statements are equivalent:

  1. contains a -profile.

  2. has a consistent -avoiding orientation (which extends ).

  3. There is no -graph over  rooted in .∎

As before, we can replace with or  ad libitum, and the -graph in (iii) can be chosen to be constructible over , or  as desired.

As in the case of -blocks, one may be tempted to compare the consistent -avoiding orientations of with those that are required only to avoid the (proper) stars in . If these are still just the -profiles, we can use the Strong Duality Theorem from [6] to characterize them by proper -trees rather than just -graphs, improving Theorem 7.4.

So, given a triple with crossing and , we can ‘uncross’ it to obtain the nested triples

As is easy to check, at least one of and must again be in  (by the ‘submodularity’ of the order of graph separations), and it would be natural to expect that the corresponding triple might then be in . If that was always the case, then every orientation of  avoiding  would in fact avoid all of . The -profiles of would then be precisely the consistent orientations of not containing a star in .

Unfortunately, however, the triple (say) can fail to lie in  for a different reason: even if the separation replacing lies in , it can happen that . We leave the details to the reader to check.

In the remainder of this section we shall see that this is not just an obstacle that might be overcome: -profiles are indeed not in general dual to -trees over stars in , and -blocks are not in general dual to -trees over stars in  (all over ).101010As noted earlier, the latter also follows indirectly from the fact that -blocks are not the same as -havens, which are dual to the -trees over stars in  [6]. The considered in [6] was slightly larger, containing also the proper separations with , but -trees over  rooted in these can easily be extended to -trees over  rooted in our .

Figure 10: The graph of Example 7.5 and its -graph
Example 7.5.

Consider the graph on a set of 12 vertices shown in Figure 10. It has five s separated in pairs by two crossing separations . The only other proper separations are of the form (or its inverse) with spanning one of s and for the ‘corner vertex’  of that .

Note that every pair of non-adjacent vertices is separated by one of these 4-separations. Hence has four 5-blocks, the s, but no 6-block. It also has no -profile. Indeed, if it did then by symmetry we could assume that this profile  contains and . By condition (P), it then also contains the 4-separation with and spanning the bottom-right  in Figure 10. (We call this a corner separation.) As , we also have