Separating layered treewidth and row treewidth

Layered treewidth and row treewidth are recently introduced graph parameters that have been key ingredients in the solution of several well-known open problems. It follows from the definitions that the layered treewidth of a graph is at most its row treewidth plus 1. Moreover, a minor-closed class has bounded layered treewidth if and only if it has bounded row treewidth. However, it has been open whether row treewidth is bounded by a function of layered treewidth. This paper answers this question in the negative. In particular, for every integer $k$ we describe a graph with layered treewidth 1 and row treewidth $k$. We also prove an analogous result for layered pathwidth and row pathwidth.


Introduction
Treewidth is a graph parameter that measures how similar a given graph is to a tree; it is of fundamental importance in structural and algorithmic graph theory; see the surveys [3,23,32].
Layered treewidth is a variant of treewidth introduced independently by Dujmović, Morin, and Wood [15] and Shahrokhi [37]; see Section 2 for the definition. A key property is that layered treewidth is bounded on planar graphs but treewidth is not. In particular, planar graphs have layered treewidth at most 3 [15], but the n × n grid graph has treewidth n. Layered treewidth has been used in upper bounds on several graph parameters including queue-number [13,15], stack number [13], boxicity [36], clustered chromatic number [28], generalised colouring numbers [26], asymptotic dimension [4], as well for results in intersection graph theory [37].
Layered and row treewidth are closely related in that the layered treewidth of a graph is at most its row treewidth plus 1, and a minor-closed class has bounded layered treewidth if and only if it has bounded row treewidth. However, a fundamental open problem is whether row treewidth is bounded by a function of layered treewidth. This paper answers this question in the negative. Theorem 1. For each k ∈ N there is a graph with layered treewidth 1 and row treewidth k.
This result is proved in Section 3. In Section 4 we use a key lemma from the proof of Theorem 1 to prove a result about the relationship between layered treewidth and queue-number.
Layered pathwidth is a graph parameter analogous to layered treewidth, first studied by Bannister, Devanny, Dujmović, Eppstein, and Wood [1] and Dujmović, Eppstein, Joret, Morin, and Wood [10]. Row pathwidth is defined in a similar way to row treewidth. We prove the following analogue of Theorem 1 for layered pathwidth and row pathwidth.
Theorem 2. For each k ∈ N there is a tree with layered pathwidth 1 and row pathwidth k.

Definitions
We consider finite, undirected, simple graphs and use standard graph theory terminology [8].

Minors
A graph H is a minor of a graph G if a graph isomorphic to H can be obtained from a subgraph of G by contracting edges. A graph class G is minor-closed if for every graph G ∈ G, every minor of G is in G. A graph class G is proper if some graph is not in G. A K t model in a graph G is a set {X 1 , . . . , X t } of pairwise-disjoint connected subgraphs in G, such that there is an edge of G between X i and X j for all distinct i, j ∈ {1, . . . , t}. Clearly, K t is a minor of G if and only if G contains a K t model.

Treewidth and Pathwidth
A tree decomposition T of a graph G is a collection (B x : x ∈ V (T )) of subsets of V (G) called bags indexed by the nodes of a tree T such that (i) for each v ∈ V (G), the induced subgraph T [{x ∈ V (T ) : v ∈ B x }] is nonempty and connected; and (ii) for each edge vw ∈ E(G), there exists x ∈ V (T ) such that {v, w} ⊆ B x . The width of a tree-decomposition is the size of its largest bag, minus 1. The treewidth tw(G) of a graph G is the minimum width of any tree-decomposition of G. If P = (B x : x ∈ V (P )) is a tree-decomposition of G and P is a path, then P is a path-decomposition of G. The pathwidth pw(G) of a graph G is the minimum width of any path-decomposition of G. For each k ∈ N, the graphs with treewidth at most k form a minor-closed class, as do the graphs with pathwidth at most k. Note that tw(K n ) = pw(K n ) = n − 1.

Layered Treewidth and Pathwidth
A layering L of a graph G is a partition of V (G) into a sequence of sets (L 0 , L 1 , . . .) such that for any edge vw ∈ E(G), if v ∈ L i and w ∈ L j then |i − j| 1. For example, if r is a vertex in a connected graph G, and L i is the set of vertices at distance i from r in G, then (L 0 , L 1 , . . . ) is the breadth-first layering of G rooted at r. A layered tree-decomposition of a graph G consists of a pair (L, T ) where L = (L 0 , L 1 , . . .) is a layering of G and T = (B x : x ∈ V (T )) is a tree-decomposition of G. The (layered) width of (L, T ) is the size of the largest intersection between a layer and a bag; that is, The layered treewidth ltw(G) of G is the minimum width of any layered tree-decomposition of G.
This definition was introduced independently by Shahrokhi [37] and Dujmović et al. [15]. The latter authors proved that ltw(G) 3 for every planar graph G; more generally that ltw(G) 2g + 3 for every graph G of Euler genus g; and that a minor-closed class G has bounded layered treewidth if and only if some apex graph (ii) is not in G. For an arbitrary proper minor-closed class G, Dujmović et al. [15] showed that every graph in G has a tree-decomposition in which each bag has a bounded set of vertices whose deletion leaves a subgraph with bounded layered treewidth. This version of Robertson and Seymour's Graph Minor Structure Theorem has proved to be very useful [4,[27][28][29].
If (L, P) is a layered tree-decomposition of G and P is a path-decomposition, then (L, P) is a layered path-decomposition of G. The layered pathwidth of G is the minimum width of any layered pathdecomposition of G. This parameter was introduced by Bannister et al. [1], who proved that every outerplanar graph has layered pathwidth at most 2 (amongst other examples).

Row Treewidth and Pathwidth
The cartesian product of graphs A and B, denoted by A B, is the graph with vertex set xy ∈ E(B). The strong product of A and B, denoted by A ⊠ B, is the union of A B and A × B.
The row treewidth rtw(G) of a graph G is the minimum treewidth of a graph H such that G is isomorphic to a subgraph of H ⊠ P ∞ , where P ∞ is the 1-way infinite path. This definition is implicit in the work of Dujmović et al. [14] who proved that rtw(G) 8 for every planar graph G; more generally, that rtw(G) 2g + 9 for every graph G of Euler genus g; and that a minor-closed class G has bounded row treewidth if and only if some apex graph is not in G. For an arbitrary minor-closed class G, Dujmović et al. [14] showed that every graph in G has a tree-decomposition in which each bag has a bounded set of vertices whose deletion leaves a subgraph with bounded row treewidth.
It follows from the definitions that for every graph G, To see this, suppose that Define the row pathwidth rpw(G) of a graph G to be the minimum pathwidth of a graph H such that G is isomorphic to a subgraph of H ⊠ P ∞ . Then lpw(G) rpw(G) + 1 for every graph G. As explained above, "bounded layered treewidth" and "bounded row treewidth" coincide for minor-closed classes. However, this is not the case for their pathwidth analogues. Dujmović et al. [10] proved that a minor-closed class G has bounded layered pathwidth if and only if some apex-forest is not in G. However, forests are a minor-closed class excluding some apex-forest (namely, K 3 ), but forests have unbounded row pathwidth by Theorem 2. In fact, Robertson and Seymour [33] proved that for every fixed forest T , the class of T -minor-free graphs has bounded pathwidth (see [2] for a tight bound). It follows that a minor-closed class G has bounded row pathwidth if and only if G has bounded pathwidth if and only if some tree is not in G.

Separating Layered Treewidth and Row Treewidth
This section proves Theorem 1. Subdivisions play a key role. A subdivision of a graph G is a graph G ′ obtained by replacing each edge vw of G with a path P vw from v to w whose internal vertices have degree 2. If each P vw has exactly s internal vertices, then G ′ is the s-subdivision of G. If each P vw has at most s internal vertices, then G ′ is a ( s)-subdivision of G. It is well known and easily proved that The proof of Theorem 1 is based on two lemmas. The first shows that subdivisions efficiently reduce layered treewidth (Lemma 3). The second shows that subdivisions do not efficiently reduce row treewidth (Lemma 6). The theorem quickly follows.

Subdivisions Efficiently Reduce Layered Treewidth
First we prove (a). We may assume that B Let G ′ be obtained from G as follows. Consider each edge e = vw of G. Say v ∈ V i and w ∈ V i ′ , and v ∈ L a and w ∈ L a ′ . Without loss of generality, a < a ′ . Then a ′ k(i ′ + 1) − 1 k(i + 2) − 1 a + 2k − 1. Replace e by the path (v, s e,a+1 , s e,a+2 , . . . , s e,a ′ −1 , w) in G ′ . The number of division vertices is a ′ − 1 − a 2k − 2. Put the division vertex s e,b in L b for each b ∈ {a + 1, . . . , a ′ − 1}. So (L 1 , L 2 , . . . ) is a layering of G ′ . Some bag B x contains v and w. Add a leaf node to T adjacent to x with corresponding bag {v, s e,a+1 , s e,a+2 , . . . , s e,a ′ −1 , w}. We obtain a tree-decomposition of G ′ with at most one vertex in each layer and in each bag. Hence ltw(G ′ ) = 1.
We now prove (b). Suppose that some ( r)-subdivision G ′ of G has ltw(G ′ ) c. Let (B x : x ∈ V (T )) be a tree-decomposition of G ′ and let (V 0 , V 1 , . . . ) be a layering of G ′ such that |B x ∩ V i | c for each x ∈ V (T ) and i ∈ N 0 . Orient each edge of G arbitrarily. For each oriented edge vw of G and for each division vertex z of vw, let α(z) := v. For each node x ∈ V (T ), let C x be obtained from B x by replacing each division vertex z ∈ B x by α(z). Observe that (C . Consider an edge vw of G with v ∈ V i and w ∈ V i ′ and i i ′ . Then i ′ i+r+1 since vw is subdivided at most r times. Say v ∈ L j and w ∈ L j ′ . By definition, j(r + 1) i (j + 1)(r + 1) − 1 and j ′ (r + 1) i ′ (j ′ + 1)(r + 1) − 1. Hence j ′ (r + 1) i ′ i + r + 1 (j + 1)(r + 1) − 1 + (r + 1) = (j + 1)(r + 1) + r, implying j ′ j + 1. That is, (L 0 , L 1 , . . . ) is a layering of G. Each layer L j contains at most c(r + 1) vertices in each bag C x . Thus k = ltw(G) c(r + 1), implying r k/c − 1.
Note that the proof of Lemma 3(a) is easily adapted to show that if s e 2k − 2 for each edge e ∈ E(G) and G ′ is the subdivision of G in which each edge e is subdivided s e times, then ltw(G ′ ) 2. We omit these straightforward details.
Proof: Let (L, T ) be a layered tree-decomposition of G ′ . Since K k+1 has radius 1, G ′ has radius at most s + 1 and therefore |L| 2s + 3. Since K k+1 has treewidth k, so does G ′ . Thus T has at least one bag B x of size at least k + 1. By the pigeonhole principle, |B x ∩ L| (k + 1)/|L| (k + 1)/(2s + 3) for at least one L ∈ L.

Subdivisions do not Efficiently Reduce Row Treewidth
This section proves the following result.
Lemma 6. For each k ∈ N and s ∈ N 0 there exists a graph G such that for every ( s)-subdivision G ′ of G, rtw(G ′ ) k tw(G).
A weaker version (with a more indirect proof) of Lemma 6 is implied by existing results about the p-centred chromatic number χ p (G) of a graph G; see [7,9] for the definition (we will not need it). Dubois et al. [9] show that, for each p ∈ N, there exists an integer k ∈ Θ( √ p) and a treewidth-k graph G, such that if G ′ is the 6k-subdivision of G, then On the other hand, Dȩbski et al. [7] (see also [16]) show that for every graph G and p ∈ N, Thus, for the graph G ′ constructed by Dubois et al. [9], This implies that rtw(G ′ ) ∈ Ω(tw(G)/ log(tw(G))), as noted in [19]. To prove Lemma 6 it will be convenient to use the language of H-partitions from [14]. An H-partition of a graph G is a partition H = (B x : x ∈ V (H)) of V (G) indexed by the nodes of some graph H such that for any edge vw ∈ E(G), if v ∈ B x and w ∈ B y , then xy ∈ E(H) or x = y. A layered H-partition (H, L) of a graph G consists of an H-partition H and a layering L of G. The (layered) width of (H, L) is max{|B x ∩ L| : x ∈ V (H), L ∈ L}. The layered width of an H-partition H of a graph G is the minimum width, taken over all layerings L of G, of the width of (H, L). Dujmović et al. [14] observed that: 14]). For all graphs G and H, G is isomorphic to a subgraph of H ⊠ P ∞ ⊠ K w if and only if G has an H-partition of layered width at most w. In particular, rtw(G) equals the minimum treewidth of a graph H for which G has a layered H-partition of layered width 1.
This observation and the next lemma (with w = 1) implies Lemma 6. Lemma 8 generalises a result of Dujmović et al. [14] who proved the s = 0 case. Lemma 8. For all w, k ∈ N and s ∈ N 0 , there exists a graph G = G s,k,w such that tw(G) k, and for any ( s)-subdivision G ′ of G, and for any graph H and any H-partition of G ′ with layered width at most w, there is a K k+1 minor in H, implying tw(H) k and rtw(G) k.
Thus the next lemma implies Lemma 8.
Lemma 9. For all w ∈ N and k, s ∈ N 0 , there exists a graph G = G s,k,w such that tw(G) k, and for any ( s)-subdivision G ′ of G, and for any graph H and any H-partition Proof: The proof is by induction on k. For the base case with k = 0, let G s,0,w be the graph with one vertex v and no edges. Any subdivision G ′ contains a model Y 1 = G ′ [{v}] of K 1 with |Y 1 | 1 = 0(2s + 1) + 1, and this model trivially respects any H-partition of G ′ , for any graph H. Now assume k 1, and that the induction hypothesis holds for k − 1. Let Q = G s,k−1,w be the graph obtained by induction. Let N := k 2 (2s + 1)(4s + 5)w + 1. Create the graph G by starting with N disjoint copies Q 1 , . . . , Q N of Q. Next add a vertex v and, for each i ∈ {1, . . . , N } and each u ∈ V (Q i ), add N internally disjoint paths of length 2 from v to u.
First we show that tw(G) k. If k = 1 then G is a subdivided star, which has treewidth 1. Now assume that k 2. Begin with any tree-decomposition (B x : x ∈ V (T )) of Q 1 ∪ . . . ∪ Q N with width at most (k − 1). Add v to every bag B x . For all i, j ∈ {1, . . . , N } and for each vertex u ∈ V (Q i ), choose a bag B x that contains u, and attach a leaf node adjacent to x whose bag contains v, u, and the degree-2 vertex of the j-th length-2 path from v to u. Each bag of this decomposition has size at most max{3, k + 1} k + 1 and therefore tw(G) k.
Let G ′ be a ( s)-subdivision of G. Let H be a graph and let (H, L) be a layered H-partition of G ′ with width at most w, where H := (A x : x ∈ V (H)). Since the radius of G is 2, the radius of G ′ is at most 2s + 2. Therefore |L| 4s + 5, and |A x | (4s + 5)w for each x ∈ V (H).
Let z be the unique node of H such that v ∈ A z , and let Q ′ 1 , . . . , Q ′ N be the (possibly subdivided) copies of Q 1 , . . . , Q N that appear in ) defines a width-w layered H i -partition of Q ′ i (with respect to layering L).

By induction, there is a
, and for each j ∈ {1, . . . , k} we have |V (Y j )| (k − 1)(2s + 1) + 1 k(2s Since N > |F |, for each j ∈ {1, . . . , k}, for at least one of the N paths between v and y j added in the construction of G, the corresponding path in G ′ avoids F . Let P j be this path, not including y j . Let Y k+1 := {P j : j ∈ {1, . . . , k}}. So V (Y k+1 ) ∩ F = ∅. By construction, there is an edge from y j to Y k+1 for each j ∈ {1, . . . , k}. So We now prove the following strengthening of Theorem 1.
See [9,22] for other examples where O(tw(G))-subdivisions of graphs G are used to prove lower bounds.

Queue Layouts and Layered Treewidth
The queue-number qn(G) of a graph G is a well-studied graph parameter introduced by Heath, Leighton and Rosenberg [24,25]; we omit the definition since we will not need it. Queue-number can be upper bounded in terms of layered treewidth and row treewidth. In particular, Dujmović et al. [15] proved that n-vertex graphs of bounded layered treewidth have queue-number in O(log n), while Dujmović et al. [14] proved that graphs of bounded row treewidth have bounded queue-number. (Row treewidth was discovered as a tool for proving that planar graphs have bounded queue-number.) This is a prime example of a difference in behaviour between layered treewidth and row treewidth. Nevertheless, it is open whether graphs of bounded layered treewidth have bounded queue-number. We show that the answer to this question depends entirely on the case of layered treewidth 1.
Corollary 11. Graphs of bounded layered treewidth have bounded queue-number if and only if graphs of layered treewidth 1 have bounded queue-number.
Proof: The forward implication is immediate. Now assume that every graph of layered treewidth 1 has queue-number at most some constant c. Let G be a graph with layered treewidth k. By Lemma 3, G has a ( k)-subdivision G ′ with layered treewidth 1. So qn(G ′ ) c. Dujmović and Wood [18] proved that for every graph H and ( s)-subdivision H ′ of H, we have qn(H) ∈ O(qn(H ′ ) 2s ). This bound was improved to O(qn(H ′ ) s+1 ) in [17]. Thus qn(G) O(c k+1 ), implying that graphs of bounded layered treewidth have bounded queue-number.
This proof highlights the value of considering the behaviour of a graph parameter on subdivisions. An analogous result holds for nonrepetitive chromatic number π(G) (using a result of Nešetřil, Ossona de Mendez, and Wood [30] to bound π(H) in terms of π(H ′ )).
Corollary 12. Graphs of bounded layered treewidth have bounded nonrepetitive chromatic number if and only if graphs of layered treewidth 1 have bounded nonrepetitive chromatic number.

Separating Layered Pathwidth and Row Pathwidth
Recall that Theorem 2 asserts that for all k ∈ N there is a tree T with lpw(T ) = 1 and rpw(T ) k. We first show that this theorem, in fact, follows from results in the literature. Bannister et al. [1] noted that lpw(T ) = 1 for every tree T . Dvorák et al. [19] showed that any family G of graphs with bounded row pathwidth has polynomial growth. The family of complete binary trees does not have polynomial growth, so for each k ∈ N, there exists a complete binary tree T with rpw(T ) k. This proves Theorem 2.
We now prove the following stronger result.
Theorem 13. For every k ∈ N there exists a tree T with pw(T ) = rpw(T ) = k.
Let T d,h be the complete d-ary tree of height h. It is folklore that pw(T d,h ) = h for all d 3 (see [20,34,35]). So Observation 7 and Lemma 14 with ℓ = 3 and w = 1 implies Theorem 13.
Lemma 14. For all h ∈ N 0 and w, ℓ ∈ N there exists d ∈ N such that for every graph H and every H-partition (A x : x ∈ V (H)) of T d,h with layered width at most w, the graph H contains a subgraph isomorphic to T ℓ,h . Moreover, if r is the root vertex of T d,h and r ∈ A z , then z is the root vertex of a subgraph of H isomorphic to T ℓ,h .

Proof:
We proceed by induction on h. The case h = 0 is trivial. Now assume that h 1. We may assume that the number of layers is at most the diameter of T d,h plus 1, which equals 2h+1. So |A x | w(2h+1) for each x ∈ V (H). Let r 1 , . . . , r d be the children of r in T d,h . Let T 1 , . . . , T d be the copies of T d,h−1 in T d,h , where T i is rooted at r i . At most w(2h + 1) − 1 of T 1 , . . . , T d intersect A z . Without loss of generality, T 1 , . . . , T d−w(2h+1)+1 do not intersect A z . For each i ∈ {1, . . . , d − w(2h + 1) + 1}, let z i be the vertex of H such that r i ∈ A zi . Since T i ∩ A z = ∅, we have z i = z. Since rr i ∈ E(T d,h ), we have zz i ∈ E(H). By induction, H − z contains a subgraph S i isomorphic to T ℓ,h−1 rooted at z i . Let X be the intersection graph of S 1 , . . . , S d−w(2h+1)+1 . If S i and S j intersect in node y of H, then T i and T j both intersect A y . Since |V (S i )| (ℓ + 1) h−1 and |A y | w(2h + 1), X has maximum degree ∆(X) w(2h + 1)(ℓ + 1) h . Thus χ(X) w(2h + 1)(ℓ + 1) h + 1. For sufficiently large d, we have |V (X)| > (ℓ − 1)χ(X). Thus, in any χ(X)-colouring of X, some colour class has at least ℓ vertices. Without loss of generality, S 1 , . . . , S ℓ are pairwise-disjoint. Hence, S 1 ∪ · · · ∪ S ℓ along with z forms a subgraph of H isomorphic to T ℓ,h , rooted at z, as desired.