# Minor-closed graph classes with bounded layered pathwidth

We prove that a minor-closed class of graphs has bounded layered pathwidth if and only if some apex-forest is not in the class. This generalises a theorem of Robertson and Seymour, which says that a minor-closed class of graphs has bounded pathwidth if and only if some forest is not in the class.

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## 1 Introduction

Pathwidth and treewidth are graph parameters that respectively measure how similar a given graph is to a path or a tree. These parameters are of fundamental importance in structural graph theory, especially in Roberston and Seymour’s graph minors series. They also have numerous applications in algorithmic graph theory. Indeed, many NP-complete problems are solvable in polynomial time on graphs of bounded treewidth [20].

Recently, Dujmović et al. [16] introduced the notion of layered treewidth. Loosely speaking, a graph has bounded layered treewidth if it has a tree decomposition and a layering such that each bag of the tree decomposition contains a bounded number of vertices in each layer (defined formally below). This definition is interesting since several natural graph classes, such as planar graphs, that have unbounded treewidth have bounded layered treewidth. Bannister et al. [1] introduced layered pathwidth, which is analogous to layered treewidth where the tree decomposition is required to be a path decomposition.

The purpose of this paper is to characterise the minor-closed graph classes with bounded layered pathwidth.

### 1.1 Definitions

Before continuing, we define the above notions. A tree decomposition of a graph is a collection of subsets of (called bags) indexed by the nodes of a tree , such that:

1. for every edge of , some bag contains both and , and

2. for every vertex of , the set induces a non-empty connected subtree of .

The width of a tree decomposition is the size of the largest bag minus 1. The treewidth of a graph , denoted by , is the minimum width of a tree decomposition of .

A path decomposition is a tree decomposition in which the underlying tree is a path. We denote a path decomposition by the corresponding sequence of bags . The pathwidth of , denoted by , is the minimum width of a path decomposition of .

A graph is a minor of a graph if a graph isomorphic to can be obtained from a subgraph of by contracting edges. A class of graphs  is minor-closed if for every , every minor of is in .

A layering of a graph is a partition of such that for every edge , if and then . Each set is called a layer. For example, for a vertex of a connected graph , if is the set of vertices at distance from , then is a layering of , called the bfs layering of starting from .

Dujmović et al. [16] introduced the following definition. The layered width of a tree decomposition of a graph is the minimum integer such that, for some layering of , each bag contains at most vertices in each layer . The layered treewidth of a graph , denoted by , is the minimum layered width of a tree decomposition of . Bannister et al. [1] defined the layered pathwidth of a graph , denoted by , to be the minimum layered width of a path decomposition of .

### 1.2 Examples and Applications

Several interesting graph classes have bounded layered treewidth (despite having unbounded treewidth). For example, Dujmović et al. [16] proved that every planar graph has layered treewidth at most 3, and more generally that every graph with Euler genus has layered treewidth at most . Note that layered treewidth and layered pathwidth are not minor-closed parameters (unlike treewidth and pathwidth). In fact, several graph classes that contain arbitrarily large clique minors have bounded layered treewidth or bounded layered pathwidth. For example, every graph that can be drawn on a surface of Euler genus with at most crossings per edge has layered treewidth at most . Even with and , this family includes graphs with arbitrarily large clique minors. Map graphs have similar behaviour [14].

Bannister et al. [1] identified the following natural graph classes that have bounded layered pathwidth (despite having unbounded pathwidth): every squaregraph has layered pathwidth 1; every bipartite outerplanar graph has layered pathwidth 1; every outerplanar graph has layered pathwidth at most 2; every Halin graph has layered pathwidth at most 2; and every unit disc graph with clique number has layered pathwidth at most .

Part of the motivation for studying graphs with bounded layered treewidth or pathwidth is that such graphs have several desirable properties. For example, Norin proved that every -vertex graph with layered treewidth has treewidth less than (see [16]). This leads to a very simple proof of the Lipton-Tarjan separator theorem. A standard trick leads to an upper bound of on the pathwidth (see [14]).

Another application is to stack layouts (or book embeddings), queue layouts and track layouts. Dujmović et al. [16] proved that every -vertex graph with layered treewidth has track- and queue-number . This leads to the best known bounds on the track- and queue-number of several natural graph classes. For graphs with bounded layered pathwidth, the dependence on can be eliminated: Bannister et al. [1] proved that every graph with layered pathwidth has track- and queue-number at most . Similarly, Dujmović et al. [17] proved that every graph with layered pathwidth has stack-number at most .

Graph colouring is another application area for layered treewidth. Esperet and Joret [19] proved that every graph with maximum degree and Euler genus is (improperly) 3-colourable with bounded clustering, which means that each monochromatic component has size bounded by some function of and . This resolved an old open problem even in the planar case . The clustering function proved by Esperet and Joret [19] is roughly . While Esperet and Joret [19] made no effort to reduce this function, their method will not lead to a sub-exponential clustering bound. On the other hand, Liu and Wood [27] recently proved that every graph with layered treewidth and maximum degree is 3-colourable with clustering . In particular, every graph with Euler genus and maximum degree is 3-colorable with clustering . This greatly improves upon the clustering bound of Esperet and Joret [19]. Moreover, the proof by Liu and Wood [27] is relatively simple, avoiding many technicalities that arise when dealing with graph embeddings. This result highlights the utility of layered treewidth as a general tool.

### 1.3 Characterisations

We now turn to the question of characterising those minor-closed classes that have bounded treewidth. The key example is the grid graph, which has treewidth . Indeed, Robertson and Seymour [31] proved that every graph with sufficiently large treewidth contains the grid as a minor. The next theorem follows since every planar graph is a minor of some grid graph. Several subsequent works have improved the bounds [32, 26, 13, 7, 6].

###### Theorem 1 (Robertson and Seymour [31]).

A minor-closed class has bounded treewidth if and only if some planar graph is not in the class.

An analogous result for pathwidth holds, where the complete binary tree is the key example (the analogue of grid graphs for treewidth). Let be the complete binary tree of height . It is well known and easily proved that , and every forest is a minor of some complete binary tree. Robertson and Seymour [30] proved the following characterisation.

###### Theorem 2 (Robertson and Seymour [30]).

A minor-closed class has bounded pathwidth if and only if some forest is not in the class.

Note that Bienstock et al. [2] proved the following quantitatively stronger result: for every forest with every graph containing no minor has pathwidth at most .

Now consider layered analogues of Theorems 2 and 1. A graph is apex if is planar for some vertex . Define the pyramid to be the apex graph obtained from the grid by adding one dominant vertex . (Here a vertex is dominant if it is adjacent to every other vertex in the graph.) The pyramid has treewidth and layered treewidth at least , since every layering uses at most three layers. Pyramids are ‘universal’ apex graphs, in the sense that every apex graph is a minor of some pyramid graph (since every planar graph is a minor of some grid graph). Dujmović et al. [16] proved the following characterisation.

###### Theorem 3 (Dujmović et al. [16]).

A minor-closed class has bounded layered treewidth if and only if some apex graph is not in the class.

Theorem 3 generalises the above-mentioned result that graphs of bounded Euler genus have bounded layered treewidth. Note that the proof of Theorem 3 uses the graph minor structure theorem and thus relies on Theorem 1.

A graph is an apex-forest if is a forest for some vertex . The following theorem is the main result of this paper.

###### Theorem 4.

A minor-closed class has bounded layered pathwidth if and only if some apex-forest is not in the class.

Theorem 4 is analogous to Theorem 3 for layered treewidth. However, unlike the proof of Theorem 3 which depends on Theorem 1, our proof of Theorem 4 does not depend on Theorem 2. In fact, Theorem 4 implies Theorem 2, as we now explain. Let be a forest, and let be a graph with no minor. Let be the apex-forest obtained from by adding a dominant vertex . Let be the graph obtained from by adding a dominant vertex . Suppose for the sake of contradiction that contains a -minor. A -minor in can be described by a mapping from the vertices of to vertex-disjoint trees in such that whenever two vertices in are adjacent, the corresponding two trees induce a connected subgraph of . From this mapping, remove two (not necessarily distinct) trees, the image of and the tree (if it exists) that contains . If the tree that contains was the image of a vertex in , then instead map to the tree that was the image of . The resulting mapping describes a -minor in , as claimed. This contradiction shows that is -minor-free. By Theorem 4, has layered pathwidth at most . Since has radius 1, at most three layers are used. Thus and have pathwidth less than .

Layered treewidth is closely related to the notion of ‘local treewidth’, which was first introduced by Eppstein [18] under the guise of the ‘treewidth-diameter’ property. A graph class has bounded local treewidth if there is a function such that for every graph in , for every vertex of and for every integer , the subgraph of induced by the vertices at distance at most from has treewidth at most . If is a linear function, then has linear local treewidth. See [21, 9, 10, 18, 20] for results and algorithmic applications of local treewidth. Dujmović et al. [16] observed that if some class has bounded layered treewidth, then has linear local treewidth. On the other hand, bounded layered treewidth is a stronger property that bounded or linear local treewidth.

Local pathwidth is defined similarly to local treewidth. A graph class has bounded local pathwidth if there is a function such that for every graph in , for every vertex of and for every integer , the subgraph of induced by the vertices at distance at most from has pathwidth at most . The observation of Dujmović et al. [16] extends to the setting of local pathwidth; see Lemma 9 below.

Theorem 4 is extended to capture local pathwidth by the following theorem, which also provides a structural description in terms of a tree decomposition with certain properties that we now introduce. If is a tree indexing a tree decomposition of a graph , then for each vertex of , let denote the subtree of induced by those nodes corresponding to bags that contain . Thus is non-empty and connected. Say that a tree decomposition of a graph is -good if its width is at most and, for every , the subtree has pathwidth at most . We illustrate this definition with two examples. Let be a tree, rooted at some vertex. For each node of , introduce a bag consisting of and its parent node (or just if is the root). Then is a tree decomposition of with width 1. Moreover, for each vertex , the subtree is a star, which has pathwidth 1. Thus every tree has a -good tree decomposition. Now, consider an outerplanar triangulation . Let be the weak dual tree (ignoring the outerface). For each node of , let be the set of three vertices on the face corresponding to . Then is a tree decomposition of with width 2. Moreover, for each vertex of , the subtree is a path, which has pathwidth 1. Thus every outerplanar graph has a -good tree decomposition (since every outerplanar graph is a subgraph of an outerplanar triangulation). These constructions are generalised via the following theorem, which immediately implies Theorem 4.

###### Theorem 5.

The following are equivalent for a minor-closed class :

1. some apex-forest graph is not in ,

2.  has bounded local pathwidth,

3.  has linear local pathwidth.

4.  has bounded layered pathwidth,

5. there exist integers and , such that every graph in  has a -good tree decomposition.

Here is some intuition about property (5). Suppose that  excludes some apex-forest graph as a minor. Since every apex-forest graph is planar, by Theorem 1, the graphs in  have bounded treewidth. Thus we should expect that the tree decompositions in (5) have bounded width. Moreover, if  has bounded layered pathwidth, then has bounded pathwidth for each vertex in each graph . Property (5) takes this idea further, and says that each subtree has bounded pathwidth, which implies that has bounded pathwidth (since the width of the tree decomposition is bounded).

In Section 2 we prove . In Section 3 we close the loop by proving . This proof uses a recent characterisation by Dang [8] of the unavoidable minors in 3-connected graphs of large pathwidth.

Throughout the proof we use the following ‘universal’ apex-forest graph. Let be the graph obtained from the complete binary tree by adding one dominant vertex. Note that and the layered pathwidth of is at least , since every layering of uses at most three layers. Since every forest is a minor of some complete binary tree, every apex-forest graph is a minor of some .

## 2 Downward Implications

###### Lemma 6.

If a graph has a tree decomposition of width indexed by a tree of pathwidth , then has pathwidth at most .

###### Proof.

Let be a tree decomposition of of width . Let be a path decomposition of of width . For , let . Then is a path decomposition of of width (since and ). ∎

###### Lemma 7.

Let and be subtrees of a tree , such that . Then

 pw(T)+1⩽(pw(T_1)+1)+(pw(T_2)+1).
###### Proof.

Let be a path decomposition of with bag size at most . Each component of is contained in and therefore has a path decomposition with bag size at most . For each such component of , there is exactly one vertex in adjacent to some vertex in (otherwise would contain a cycle consisting of two edges between a path in and a path in ). Say is in bag . We say attaches at and at . By doubling bags in the path decomposition of , we may assume that distinct components of attach at distinct . For each component of , if is a path decomposition of with bag size at most , then replace by . We obtain a path decomposition of with bag size at most . The result follows. ∎

###### Corollary 8.

Let be subtrees of a tree , such that . Then

 pw(T)+1⩽∑_i=1k(pw(T_i)+1).

We now prove the downward implications in Theorem 5. First note that (2) implies (1), since if every graph in  has local pathwidth at most , then the apex-forest graph is not in . It is immediate that (3) implies (2). That (4) implies (3) is the above-mentioned observation of Dujmović et al. [16] specialised for pathwidth. We include the proof for completeness.

###### Lemma 9.

Let  be a class of graphs such that every graph in  has layered pathwidth at most . Then  has linear local pathwidth with binding function .

###### Proof.

For a graph , let be a path decomposition of with layered width , with respect to some layering . Let be a vertex in . Let be a positive integer. Let be the subgraph of induced by the vertices at distance at most from . Thus . Each bag contains at most vertices in each layer. Hence is a path decomposition of with at most vertices in each bag. Therefore  has linear local pathwidth with binding function . ∎

The next lemma shows that (5) implies (4).

###### Lemma 10.

If a graph has a -good tree decomposition, then .

###### Proof.

Let be a tree decomposition of with width , such that for each vertex of . Since adding edges does not decrease the layered pathwidth, we may add edges to between two non-adjacent vertices in the same bag of . Now each bag is a clique, and is chordal with maximum clique size . Let be a bfs layering in . That is, is the set of vertices in at distance from some fixed vertex of . In particular, .

Consider a component of for some . Let be the set of vertices in adjacent to at least one vertex in . Since is chordal, is a clique of size at most (see [25, 15]), called the parent clique of . Define . Since is a clique, which is contained in a single bag of , there is a node of such that for each . Thus is a (connected) subtree of . Moreover, is the union of at most subtrees, each with pathwidth at most . Thus by Corollary 8. Let .

We now prove that is a tree decomposition of . We first prove condition (ii). For a vertex of , the set of bags of that contain is precisely those indexed by nodes in , which is non-empty and connected, by assumption. Now, consider a vertex in . Let be the neighbour of on a shortest -path in . Thus is in . Since is an edge, and appear in a common bag of , which corresponds to a node in (since that bag contains ). Hence is non-empty. We now prove that is connected. Let and be distinct bags of containing . Let be the -path in . Since is connected, is in the bag associated with each node in . To conclude that is connected, it remains to prove that . By construction, some vertex is in and some vertex is in . Since and are adjacent, the bag associated with each node in contains or . Hence and is connected. This proves condition (ii). Now we prove condition (i). Since is contained in some bag of , condition (i) holds for each edge with endpoints in . For each edge with and , and are in a common bag of , implying is in (since contains ), as desired. Finally, consider an edge with . Suppose on the contrary that and have no common neighbour in . By construction, has a neighbour in , and has a neighbour in . Thus . Since is a clique, and are adjacent. Since and , the 4-cycle is chordless, and is not chordal, which is a contradiction. Hence and have a common neighbour in . Thus is a triangle in , which is in a common bag of , and therefore in a common bag of , implying that and are in a common bag of . This proves condition (i) in the definition of tree decomposition. Therefore is a tree decomposition of . By construction, it has width at most .

Since and indexes a tree decomposition of with width at most , by Lemma 6, .

We now construct a path decomposition of with layered width at most with respect to layering . Let . We now prove, by induction on , that has a path decomposition with layered width at most with respect to layering . This claim is trivial for . Now assume that is a path decomposition of with layered width at most with respect to layering . For each component of , there is a bag that contains ; pick one such bag and call it the parent bag of . By doubling the bags, we may assume that distinct components of have distinct parent bags. Now, for each component of with parent bag , if is a path decomposition of with width , then replace by . Doing this for each component of produces a path decomposition of with layered width at most with respect to layering . In particular, we obtain a path decomposition of with layered width at most with respect to layering . ∎

## 3 Proof that (1) implies (5)

The goal of this section is to show that if a graph excludes some apex-forest graph as a minor, then has -good tree decomposition for some and . Since every apex-forest graph is a minor of some , it suffices to prove this result for , in which case we denote and .

We will be working with two related trees and and one graph . To help the reader keep track of things we use variables , , and as names for nodes of and and variables , , , and to refer to vertices of .

We now give an outline of the proof. First, we show that a recent result by Dang [8] implies that every 3-connected graph with no minor has pathwidth at most . Thus, in this case, has a -good tree decomposition. Next we deal with cut vertices by showing that if each block of a graph has a -good tree decomposition, then has a -good tree decomposition.

Therefore, the main difficulty is to show that every 2-connected graph with no minor has a -good tree decomposition . By the result of Bienstock et al. [2] described in the introduction, if for some then contains a minor. For sufficiently large , we then construct a minor (from the minor in ) in which plays the role of the apex vertex.

To construct the tree decomposition we use two tools: An SPQR-tree, , represents a graph as a collection of subgraphs (S- and R-nodes) that are joined at 2-vertex cutsets (P-nodes). These subgraphs consist of cycles (S-nodes) and 3-connected graphs (R-nodes). Cycles have pathwidth 2 and, by the result of Dang discussed above, the 3-connected graphs have pathwidth at most . Replacing the S- and R-nodes of the SPQR-tree with these path decompositions produces the tree in our tree decomposition.

To show that this tree decomposition is -good, we first show that if contains a subdivision of a sufficiently large complete binary tree, then the SPQR-tree also contains a subdivision of a large complete binary tree all of whose nodes have subgraphs that contain . Using this large binary tree in we then piece together a subgraph of that has a minor in which is the apex vertex.

### 3.1 Dang’s Result

First we show how the following result of Dang [8] implies that every 3-connected graph with no minor has pathwidth at most .

###### Theorem 11 (Dang [8, Theorem 1.1.5]).

Let be a graph with two distinct vertices and such that is a forest, be a graph with a vertex such that is outerplanar, and be a tree with a cycle going through its leaves in order from the leftmost leaf to the rightmost leaf so that is planar. Then there exists a number such that every 3-connected graph of pathwidth at least has a , , or minor.

Note that is a Halin graph, except that degree-2 vertices are allowed in the tree.

To use Theorem 11 we need a small helper lemma. For every , let be the graph obtained from the complete binary tree of height by adding a new vertex adjacent to the leaves. The next lemma is well known.

###### Lemma 12.

For every integer , contains as a minor.

###### Proof.

The statement is immediate for . For , partition the edges of into the tree and the remaining edges, which form a star centered at some vertex . Let be the grandchildren of the root of ordered from left to right. Contract the entire subtree comprised of the subtree rooted at , the subtree rooted at , and the path from to . Applying the same procedure recursively on the copy of rooted at and the copy of rooted at produces , as can be easily verified by induction. ∎

###### Corollary 13.

There exists a number such that every 3-connected graph of pathwidth at least has a minor.

###### Proof.

Let be obtained from the complete binary tree by adding two dominant vertices and . Let be the graph obtained from the outerplanar graph , whose weak dual is a complete binary tree of height , by adding a dominant vertex . Let be the graph obtained from by adding a cycle on its leaves, so that is planar.

Then contains as a minor since is isomorphic to . also contains as a minor because contains a complete binary tree of height as a subgraph. Finally, also contains a minor: Contract the cycle, then we have a complete binary tree of height plus an apex vertex linked to its leaves, which contains as a minor by Lemma 12. Theorem 11 implies that there exists such that every 3-connected graph with pathwidth at least contains at least one of , , or as a minor and therefore contains a minor. ∎

### 3.2 Dealing with Cut Vertices

A block in a graph is either a maximal 2-connected subgraph, the subgraph induced by the endpoints of a bridge edge, or the subgraph induced by an isolated vertex.

###### Lemma 14.

Let be a graph, such that each block of has a -good tree decomposition. Then has a -good tree decomposition.

###### Proof.

Let be the blocks of . For each , let be the underlying tree in a -good tree decompositions of .

We create a tree decomposition of as follows: For each cut vertex or isolated vertex in , introduce a new tree node with . In each block that contains , the tree decomposition of has at least one node such that ; make adjacent to exactly one such node for each .

It is straightforward to verify that this defines a tree decomposition of and we now argue this decomposition is -good. The resulting tree decomposition of has width at most . For each isolated vertex , the subtree consists of one node. For each cut vertex , the subtree is composed of some number of subtrees, each adjacent to and each having a path decomposition of width at most . We obtain a path decomposition of by concatenating the path decompositions of each subtree and adding to every bag of the resulting path decomposition. The resulting path decomposition of has width at most . ∎

### 3.3 SPQR-Trees

In this section, we quickly review SPQR-trees, a structural decomposition of 2-connected graphs used previously to characterize planar embeddings [28], to design efficient algorithms for triconnected components [23], and in efficient data structures for incremental planarity testing [11, 12].

Let be a 2-connected graph. An SPQR-tree of is a tree in which each node is associated with a minor of . For any S- or R-node of , is a simple graph. If is a P-node, on the other hand, then is a dipole graph having two vertices and at least two parallel edges. In all cases, is a minor of . For a P-node in which contains vertices and and parallel edges, this means that contains internally disjoint paths from to . For each node of each edge

is classified either as a

virtual edge or a real edge. An SPQR-tree is defined recursively as follows (see Figure 1):111This definition includes P-nodes consisting of only two virtual edges, which some works exclude because they are unnecessary. However, their inclusion simplifies some of our analysis.

1. If is a cycle, then consists of a single node (an S-node) in which and all edges of are real.

2. If is 3-connected, then consists of a single node (an R-node) in which and all edges of are real.

3. Otherwise has a cutset such that and each have degree at least 3. Then let , , be the connected components of . For each , let be along with the additional edge , if not already present. Note that (because of the inclusion of ) each is 2-connected, so each has an SPQR-tree . Then an SPQR-tree for is obtained by creating a node (a P-node) with being a dipole graph with vertices and and having virtual edges joining and . In addition to these virtual edges, contains the real edge if . The construction and the fact that is an edge in each imply that, for each , there exists exactly one node in such that is a real edge in . To complete , make adjacent to each of , and make a virtual edge in each of .

Let be an SPQR-tree of a 2-connected graph . For each node of , we let denote the set of real edges in and denote the multiset of virtual edges in . For a connected subtree of we define as the subgraph of whose vertex set is and whose edge set is . For a vertex , let , which is called the subtree of induced by . We make use of the following properties of :

1. Every R-node and S-node is adjacent only to P-nodes and no two P-nodes are adjacent.

2. The degree of every node is equal to the number of virtual edges in .

3. For every vertex , is connected.

4. If is an R-node or S-node, then is a simple graph; that is, contains no parallel edges.

5. If a P-node has degree 2 and both its neighbors are S-nodes then has a real edge.

6. For each node of and component of , is connected.

7. For each there is exactly one node of for which is a real edge in .

### 3.4 The Good Tree Decomposition

To obtain our good tree decomposition of a 2-connected graph we start with an SPQR-tree for . For each R-node or S-node of , let be minimum-width path decomposition of . We say that the node generates the nodes in the path and that each node in is generated by .

Each S- or R-node is adjacent to some set of P-nodes in . For each such P-node whose dipole graph has vertices and , the edge is a (virtual) edge in and therefore and appear in some common bag with . We make and adjacent in . This defines the tree in the tree decomposition.

We now describe the contents of ’s bags. Each P-node of becomes a node in whose bag contains only the two vertices of . Every node in that is generated by an S- or R-node of is a node in some path decomposition of and already has an associated bag that it inherits from this path decomposition.

It is straightforward to verify that is indeed a tree decomposition of : For each vertex , the connectivity of the subtree follows from Property 3 of SPQR-trees and the equivalent property for the path decompositions that include . Each edge of appears as an edge in for at least one node of and therefore and appear in a common bag in the path decomposition of .

Each bag of either has size in (when is generated by a P-node or an S-node) or it has size at most where is the function in Corollary 13 (when is generated by an R-node). Thus, is a tree decomposition of whose width is upper bounded by a function of . It remains to show that, for every , has pathwidth that is upper bounded by a function of .

In the remainder of this section, we fix to be a 2-connected graph, to be an SPQR-tree of , and to be a tree decomposition of obtained using the procedure described above.

###### Lemma 15.

For every integer , if has pathwidth greater than , then contains a subdivision of .

###### Proof.

In the following, a binary tree is a tree rooted at a degree-2 node, such that every other node has degree in . In a binary tree, the root and every degree 3 node is called a branching node. Every branching node and every leaf is a distinctive node. We use the convention that all binary trees are ordered, possibly arbitrarily, so that we can distinguish between the left and right child of a branching node. For a node in a binary tree , we denote by the subtree rooted at ; that is, the subtree of induced by the set of nodes that have as an ancestor, including itself.

Recall, from the result of Bienstock et al. [2] discussed in the introduction, that if has pathwidth greater than then contains a subdivision of . Note that does not immediately imply the existence of in since two or more distinctive nodes of may have been generated by the same node of . Label each node of with the node of that generated it. Recall that each node in generates a path in . So a maximal subset of nodes of with a common label induces a path in .

We claim that contains a subdivision of such that each of the distinctive nodes of has a unique label. We establish this claim by induction on : If then the claim is trivial. Otherwise, let be the root of and let and be the highest branching nodes in the left and right subtrees of , respectively. Let and be the highest distinctive nodes in the left and right subtrees of , respectively, and let and be the highest distinctive nodes in the left and right subtrees of , respectively. Since each label induces a path in , at least one of , say , and at least one of , say , does not have the same label as . Furthermore, since and are separated by , the set of labels of nodes in is disjoint from the set of labels of nodes in . Applying induction on and yields two subdivisions of in which each distinctive node has a unique label. Connecting these two subdivisions with the unique path from to yields the desired subdivision of in which each distinctive node has a unique label.

Now, each distinctive node in has a unique label and therefore corresponds to a unique node of . Thus, if we contract all nodes of sharing a common label, then we obtain a subtree of that is a subdivision of . ∎

Thus far we have established that if has sufficiently high pathwidth, then contains a subdivision of a large complete binary tree.

###### Lemma 16.

If contains a subdivision of then contains a minor.

###### Proof.

First we note that if contains a subdivision of then contains a subdivision of such that the path between each pair of distinctive nodes in has length at least 7.

It is convenient to work with a simplified SPQR-tree and graph obtained by repeating the following operation exhaustively: Consider some edge of with and . The edge is associated with some virtual edge in . In , replace the virtual edge in with a real edge. At the same time, remove the maximal subtree of that contains and not . By Property 6 of SPQR trees, in this operation is equivalent to contracting all the real edges in and removing any resulting parallel edges. Since the resulting graph is a minor of , this operation is safe in the sense that the existence of a minor in implies the existence of a minor in .

With this simplification, the tree is an SPQR-tree for the graph and every virtual edge is incident to . We now turn our efforts to finding the minor in . Recall that is obtained from a complete binary tree by adding a dominant vertex. We begin by finding a subdivision of in . In this subdivision, each edge of that joins a node to its left child is represented by a path joining a branching node to a distinctive node . We show that contains a path from to some anchor node of with , which is vertex disjoint from except for . Furthermore, except for their common endpoint , all of these paths are disjoint. The union of and these paths contains a minor since contracting the path from each anchor node to its closest ancestor branching node produces . See Figure 2.

Let be a branching node of and let be the nearest distinctive node in one of ’s two subtrees. Consider the path in . For each , the edge is associated with a cutset in and is a virtual edge in and . Note that this implies that, for each , contains both vertices and .

We claim that, for each , contains a path from to that does not contain ; refer to Figure 3. When is a P-node, this claim is trivial since, in this case, . The case in which is an S-node or R-node is also easy: In these cases is 2-connected, therefore there is a path from to that avoids . Now note that the paths are disjoint, except for each of the common endpoints where ends and begins. This is because each is a cutset of that separates from . By concatenating we obtain a path from to that we call the subdivision path for nodes and .

By Properties 2 and 4 of SPQR-trees and the fact that all virtual edges are incident to , every branching node of is either an R-node or a P-node. In the case where is a P-node, all the subdivision paths that begin or end at a vertex of include the same vertex of . In the case where is an R-node, each subdivision path that begins or ends at a vertex of includes a different vertex (for up to 3 different vertices , , and ). Now, since is 3-connected, these three vertices are in the same component of . In particular, contains an edge-minimal tree that includes , , and . Adding each of these trees to the union of all subdivision paths produces the subdivision of .

Next we show how to construct paths from to anchor nodes. Let be a branching node of , let be the highest distinctive node in ’s left subtree and let be the path in having endpoints and . Thus far we have established that contains a simple path from to that does not include . We now show that contains an