Table Of ContentOrthogonal Tree Decompositions of Graphs
Vida Dujmovi´c∗ Gwena¨el Joret† Pat Morin‡ Sergey Norin§ David R. Wood¶
Abstract. This paper studies graphs that have two tree decompositions with the property
that every bag from the first decomposition has a bounded-size intersection with every
bag from the second decomposition. We show that every graph in each of the following
7 classes has a tree decomposition and a linear-sized path decomposition with bounded
1
intersections: (1) every proper minor-closed class, (2) string graphs with a linear number
0
2 of crossings in a fixed surface, (3) graphs with linear crossing number in a fixed surface.
n
We then show that every n-vertex graph that has a tree decomposition and a linear-sized
a
√
J path decomposition with bounded intersections has O( n) treewidth. As a corollary, we
9
conclude a new lower bound on the crossing number of a graph in terms of its treewidth.
1
Finally, we consider graph classes that have two path decompositions with bounded
]
O intersections. Trees and outerplanar graphs have this property. But for the next most
C
simple class, series parallel graphs, we show that no such result holds.
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a January 23, 2017
∗ School of Computer Science and Electrical Engineering, University of Ottawa, Ottawa, Canada
([email protected]). Supported by NSERC.
†D´epartement d’Informatique, Universit´e Libre de Bruxelles, Belgium ([email protected]). Supported
by an Action de Recherches Concert´ees grant from the Wallonia-Brussels Federation of Belgium
‡SchoolofComputerScience,CarletonUniversity,Ottawa,Canada([email protected]). Research
supported by NSERC.
§Department of Mathematics and Statistics, McGill University, Montr´eal, Canada
([email protected]). Supported by NSERC grant 418520.
¶ School of Mathematical Sciences, Monash University, Melbourne, Australia ([email protected]).
Supported by the Australian Research Council.
1
1 Introduction
A tree decomposition represents the vertices of a graph as subtrees of a tree, so that the
subtrees corresponding to adjacent vertices intersect. The treewidth of a graph G is the
minimum taken over all tree decompositions of G, of the maximum number of subtrees
intersecting at a single node of the tree minus 1. Treewidth measures how similar a
given graph is to a tree. It is a key measure of the complexity of a graph and is of
fundamental importance in algorithmic graph theory and structural graph theory. For
example, treewidth is a key parameter in Robertson–Seymour graph minor theory [46],
and many NP-complete problems are solvable in polynomial time on graphs of bounded
treewidth [13].
The main idea in this paper is to consider two tree decompositions of a graph, and
then measure the size of the intersection of a bag from the first decomposition with
a bag from the second decomposition. One can think of the two tree decompositions
as being ‘orthogonal’ to each other. We are interested in which graphs have two tree
decompositions such that every bag from the first decomposition has a bounded-size
intersection with every bag from the second decomposition. This idea is implicit in recent
work on layered tree decompositions (see Section 2), and was made explicit in the recent
survey by Norin [42].
Grid graphs illustrate this idea well; see Figure 1. Say G is the n×n planar grid graph.
The sequence of consecutive pairs of columns determines a tree decomposition, in fact, a
path decomposition with bags of size 2n. Similarly, the sequence of consecutive pairs of
rows determines a path decomposition with bags of size 2n. Observe that the intersection
of a bag from the first decomposition with a bag from the second decomposition has size
4. It is well known [32] that G has treewidth n, which is unbounded. But as we have
shown, G has two tree decompositions with bounded intersections. This paper shows that
many interesting graph classes with unbounded treewidth have two tree decompositions
with bounded intersections (and with other useful properties too).
Before continuing, we formalise these ideas. A tree decomposition of a graph G is given
by a tree T whose nodes index a collection (T ⊆ V(G) : x ∈ V(T)) of sets of vertices
x
in G called bags, such that (1) for every edge vw of G, some bag T contains both v
x
and w, and (2) for every vertex v of G, the set {x ∈ V(T) : v ∈ T } induces a non-empty
x
(connected) subtree of T. For brevity, we say that T is a tree decomposition (with the
bags T implicit). The width of a tree decomposition T is max{|T |−1 : x ∈ V(T)},
x x
and the treewidth tw(G) of a graph G is the minimum width of the tree decompositions
2
Figure 1: Two 4-orthogonal path decompositions of the grid graph.
of G. A path decomposition is a tree decomposition in which the underlying tree is a
path. We describe a path decomposition simply by the corresponding sequence of bags.
The pathwidth pw(G) of a graph G is the minimum width of a path decomposition of G.
Two tree decompositions A and B of a graph G are c-orthogonal if |A ∩B | (cid:54) c for all
x y
x ∈ V(A) and y ∈ V(B).
It turns out that not only the size of bag intersections is important when considering
orthogonal tree decompositions. A key parameter is the total size of the bags in a
tree decomposition T, which we call the magnitude, formally defined to be (cid:80) |T |.
x∈V(T) x
For example, consider the complete bipartite graph K . Say V = {v ,...,v } and
n,n 1 n
W = {w ,...,w } are the two colour classes. Then
1 n
P = (V ∪{w },V ∪{w },...,V ∪{w }) and
1 2 n
Q = (W ∪{v },W ∪{v },...,W ∪{v })
1 2 n
are path decompositions of K , such that the intersection of each bag of P with each
n,n
bag of Q has exactly two vertices. However, both P and Q have magnitude n(n+1).
On the other hand, we prove in Section 3 that if two tree decompositions of a graph G
have bounded intersections and one has linear magnitude, then G has a linear number
of edges. Here ‘linear’ means O(n) for n-vertex graphs.
Our main results show that every graph in each of the following classes has a tree
decomposition and a linear-magnitude path decomposition with bounded intersections:
• every proper minor-closed class (Section 4),
3
• string graphs with a linear number of crossings in a fixed surface (Section 5),
• graphs with linear crossing number in a fixed surface (Section 6),
The latter two examples highlight that orthogonal decompositions are of interest well
beyond the world of minor-closed classes. We also show that every graph that has a tree
decomposition and a linear-magnitude path decomposition with bounded intersections
√
has O( n) treewidth. This result is immediately applicable to each of the above three
classes. As a corollary, we conclude a new lower bound on the crossing number of a
graph in terms of its treewidth.
Treewidth is intrinsically related to graph separators. A set S of vertices in a graph G is
a separator of G if each component of G−S has at most 1|V(G)| vertices. Graphs with
2
small treewidth have small separators, as shown by the following result of Robertson
and Seymour [47]:
Lemma 1 ([47]). Every graph G has a separator of size at most tw(G)+1.
√
Our treewidth bounds and Lemma 1 give O( n) separator results for each of the above
three classes. Also note that a converse to Lemma 1 holds: graphs in which every
subgraph has a small separator have small treewidth [22, 45].
The paper then considers graph classes that have two path decompositions with bounded
intersections. Trees and outerplanar graphs have this property. But for the next most
simple class, series parallel graphs, we show that no such result holds (Section 7). The
paper concludes by discussing connections between orthogonal tree decompositions and
boxicity (Section 8) and graph colouring (Section 9).
2 Layered Treewidth
Thestartingpointforthestudyoforthogonaltreedecompositionsisthenotionofalayered
tree decomposition, introduced independently by Dujmovi´c et al. [21] and Shahrokhi
[53]. Applications of layered treewidth include nonrepetitive graph colouring [21], queue
layouts, track layouts and 3-dimensional graph drawings [21], book embeddings [20], and
intersection graph theory [53].
A layering of a graph G is a partition (V ,V ,...,V ) of V(G) such that for every edge
0 1 t
vw ∈ E(G), if v ∈ V and w ∈ V , then |i−j| (cid:54) 1. Each set V is called a layer. For
i j i
4
example, for a vertex r of a connected graph G, if V is the set of vertices at distance i
i
from r, then (V ,V ,...) is a layering of G.
0 1
The layered width of a tree decomposition (T : x ∈ V(T)) of a graph G is the minimum
x
integer (cid:96) such that, for some layering (V ,V ,...,V ) of G, each bag T contains at most
0 1 t x
(cid:96) vertices in each layer V . The layered treewidth of a graph G is the minimum layered
i
width of a tree decomposition of G. Note that the trivial layering with all vertices in one
layer shows that layered treewidth is at most treewidth plus 1. The layered pathwidth
of a graph G is the minimum layered width of a path decomposition of G; see [3].
√
While n-vertex planar graphs may have treewidth as large as n, Dujmovi´c et al. [21]
proved the following1:
Theorem 2 ([21]). Every planar graph has layered treewidth at most 3. More generally,
every graph with Euler genus g has layered treewidth at most 2g +3.
Layered treewidth is related to local treewidth, which was first introduced by Eppstein
[24] under the guise of the ‘treewidth-diameter’ property. A graph class G has bounded
local treewidth if there is a function f such that for every graph G in G, for every vertex
v of G and for every integer r (cid:62) 0, the subgraph of G induced by the vertices at distance
at most r from v has treewidth at most f(r); see [14, 16, 24, 30]. If f(r) is a linear
function, then G has linear local treewidth. Dujmovi´c et al. [21] observed that if every
graph in some class G has layered treewidth at most k, then G has linear local treewidth
with f(r) (cid:54) k(2r+1)−1. Dujmovi´c et al. [21] also proved the following converse result
for minor-closed classes, where a graph G is apex if G−v is planar for some vertex
v. (Earlier, Eppstein [24] proved that (b) and (d) are equivalent, and Demaine and
Hajiaghayi [16] proved that (b) and (c) are equivalent.)
Theorem 3 ([16, 21, 24]). The following are equivalent for a minor-closed class G of
graphs:
(a) G has bounded layered treewidth.
(b) G has bounded local treewidth.
(c) G has linear local treewidth.
1 The Euler genus of an orientable surface with h handles is 2h. The Euler genus of a non-orientable
surface with c cross-caps is c. The Euler genus of a graph G is the minimum Euler genus of a surface in
which G embeds (with no crossings).
5
(d) G excludes some apex graph as a minor.
Dujmovi´c et al. [19] observed that such a converse result does not hold for non-minor-
closed classes. Indeed, they showed that the 3-dimensional grid graph has quadratic
local treewidth and unbounded layered treewidth.
A number of non-minor-closed classes also have bounded layered treewidth. Dujmovi´c
et al. [19] gave the following two examples. A graph is (g,k)-planar if it can be drawn in
a surface of Euler genus at most g with at most k crossings on each edge. Dujmovi´c et al.
[19] determined an optimal bound on the layered treewidth and treewidth of such graphs.
Theorem4([19]). Every(g,k)-planargraphGhaslayeredtreewidthatmost(4g+6)(k+1)
(cid:112)
and treewidth at most 2 (4g +6)(k +1)n. Conversely, for all g,k (cid:62) 0 and infinitely
(cid:112)
many n there is an n-vertex (g,k)-planar graph with treewidth Ω( (g +1)(k +1)n)
and layered treewidth Ω((g +1)(k +1)).
Map graphs are defined as follows. Start with a graph G embedded in a surface of Euler
0
genus g, with each face labelled a ‘nation’ or a ‘lake’, where each vertex of G is incident
0
with at most d nations. Define a graph G whose vertices are the nations of G , where
0
two vertices are adjacent in G if the corresponding faces in G share a vertex. Then G
0
is called a (g,d)-map graph. A (0,d)-map graph is called a (plane) d-map graph; such
graphs have been extensively studied [10–12, 15, 26]. It is easily seen that (g,3)-map
graphs are precisely the graphs of Euler genus at most g [12, 19]. So (g,d)-map graphs
provide a natural generalisation of graphs embedded in a surface. Note that if a vertex
of G is incident with d nations, then G contains K , which is arbitrarily large. Dujmovi´c
0 d
et al. [19] determined an optimal bound on the layered treewidth and treewidth of such
graphs.
Theorem 5 ([19]). Every (g,d)-map graph on n vertices has layered treewidth at most
(cid:112)
(2g +3)(2d+1) and treewidth at most 2 (2g +3)(2d+1)n. Moreover, for all g (cid:62) 0
and d (cid:62) 8, for infinitely many integers n, there is an n-vertex (g,d)-map graph with
(cid:112)
treewidth at least Ω( (g +1)dn) and layered treewidth at least Ω((g +1)d).
Theorem 3 leads to further results. A tree decomposition is domino if every vertex is in
at most two bags [5, 6, 56].
Lemma 6. Every graph with layered treewidth k has a domino path decomposition P
and a tree decomposition T such that for every vertex v of G, the union of the bags of P
that contain v has width at most 3k −1 in T.
6
Proof. Let T be a tree decomposition of G with layered width k with respect to some
layering V ,...,V of G, where V = ∅. Then P := (V ∪V ,V ∪V ,...,V ∪V ) is a
1 t t 1 2 2 3 t−1 t
path decomposition of G. Consider a vertex v ∈ V for some i ∈ [t−1]. Then v is in
i
exactly two bags, V ∪V and V ∪V . Thus P is domino. The union of the bags that
i−1 i i i+1
contain v is V ∪V ∪V , which contains at most 3k vertices in each bag of T.
i−1 i i+1
Theorem 3 and Lemma 6 imply:
Theorem 7. For every fixed apex graph H, there is a constant k, such that every H-
minor-free graph G has a domino path decomposition P and a tree decomposition T
such that for every vertex v of G, the union of the bags of P that contain v has width at
most 3k −1 in T.
This result is best possible in the following sense. Let G be obtained from the n×n
grid graph by adding one dominant vertex v. Say T and T are tree decompositions of
1 2
G. The bags of T that contain v induce a subgraph that contains the n×n grid, and
1
therefore has treewidth at least n, which is unbounded.
3 Extremal Questions and Treewidth Bounds
We start this section by considering the natural extremal question: what is the maximum
number of edges in an n-vertex graph that has two orthogonal tree decompositions of
a particular type? Dujmovi´c et al. [21] proved that every n-vertex graph with layered
treewidth k has minimum degree at most 3k −1 and thus has at most (3k −1)n edges,
which is tight up to a lower order term. For more general structures the situation is more
delicate, since K has two 2-orthogonal path decompositions, as shown in Section 1.
n,n
Note that each of these decompositions has quadratic magnitude. We now show that a
limit on the magnitude of one decomposition leads to a linear bound on the number of
edges, even for tree decompositions.
Lemma 8. Let S and T be k-orthogonal tree decompositions of a graph G, where
S has magnitude s. Then |E(G)| (cid:54) (k − 1)s. In particular, if s (cid:54) c|V(G)| then
|E(G)| (cid:54) c(k −1)|V(G)|.
Proof. Each edge of G is in G[S ] for some x ∈ V(S). Since T restricted to G[S ] has
x x
treewidth at most k −1, it follows that G[S ] has less than (k −1)|S | edges. Thus
x x
(cid:88) (cid:88)
|E(G)| (cid:54) |E(G[S ])| (cid:54) (k −1)|S | = (k −1)s.
x x
x x
7
√
One application of layered treewidth is that it leads to O( n) treewidth bounds.
Theorem 9 (Norine; see [21]). For every n-vertex graph G with layered treewidth k,
√
tw(G) (cid:54) 2 kn−1.
As an example, Theorems 2 and 9 imply graphs with bounded Euler genus have treewidth
√
O( n). Dujmovi´c et al. [19] observed that a standard trick applied with Theorem 9
implies:
Theorem 10 ([19]). For every n-vertex graph G with layered treewidth k,
√
pw(G) (cid:54) 11 kn−1.
We now generalise these results to the setting of orthogonal decompositions. A weak
path decomposition of a graph G is a sequence P ,...,P of sets of vertices of G called
1 t
bags, such that ∪ P = V(G), for every vertex v of G the set of bags that contain v
i i
forms a subsequence, and for every edge vw of G, both v and w are in P ∪ P for
i i+1
some i ∈ {1,...,t} (where P means ∅). Note that a path decomposition is a weak
t+1
path decomposition in which the final condition is strengthened to say that both v and
w are in P for some i ∈ {1,...,t}. If P ,...,P is a weak path decomposition, then
i 1 t
P ∪P ,P ∪P ,...,P ∪P is a path decomposition with at most twice the width of
1 2 2 3 t−1 t
P ,...,P . In this sense, there is little difference between weak path decompositions and
1 t
path decompositions. Also note that a layering is a weak path decomposition in which
each vertex is in exactly one bag. Thus weak path decompositions with linear magnitude
generalise the notion of a layering. In a weak path decomposition, each bag P separates
i
P ∪···∪P and P ∪···∪P ; that is, there is no edge between P ∪···∪P and
1 i−1 i+1 t 1 i−1
P ∪···∪P . This property is the key to the next lemma, which generalises Theorem 9
i+1 t
to the setting of weak path decompositions.
Lemma 11. Suppose that T is a tree decomposition and P is a weak path decomposition
of a graph G, where T and P are k-orthogonal and P has magnitude s. Then
√
tw(G) (cid:54) 2 ks−1.
(cid:112)
Proof. Let t := (cid:100) s/k(cid:101). Label the bags of P in order by 1,...,t,1,...,t,.... Since
the magnitude of P is s, for some i ∈ {1,...,t} the bags labelled i have total size at
most s/t. Let G(cid:48) be the subgraph of G obtained by deleting the bags labelled i. Since
each bag separates the bags before and after it, each connected component of G(cid:48) is
8
contained within t−1 consecutive bags of P. Thus G(cid:48) has a tree decomposition with
bags of size at most (t−1)k. Add all the vertices in bags of P labelled i to every bag
of this tree decomposition of G(cid:48). We obtain a tree decomposition of G with bag size at
√ √
most (t−1)k +s/t (cid:54) 2 ks. Thus tw(G) (cid:54) 2 ks−1.
Several comments on Lemma 11 are in order.
First we show that Lemma 11 cannot be strengthened for two tree decompositions with
bounded intersections. Let G be a bipartite graph with bipartition (A,B) and maximum
degree ∆. Let S be the star decomposition of G with root bag A and a leaf bag N[w] for
each vertex w ∈ B. Symmetrically, let T be the star decomposition of G with root bag B
and a leaf bag N[v] for each vertex v ∈ A. Observe that S and T are ∆-orthogonal and
both have magnitude |V(G)|+|E(G)|. Now, apply this construction with G a random
cubic bipartite graph on n vertices. We obtain two 3-orthogonal tree decompositions of
G both with magnitude 5n. But it is well known that G has treewidth Ω(n); see [31]
2
for example. Thus Lemma 11 does not hold for two tree decompositions with bounded
intersections.
We now show that Lemma 11 proves that certain graph classes have bounded expansion.
A graph class C has bounded expansion if there exists a function f such that for every
graph G ∈ C, for every subgraph G(cid:48) of G, and for all pairwise disjoint balls B ,...,B of
1 s
radius at most r in G(cid:48), the graph obtained from G(cid:48) by contracting each B into a vertex
i
has average degree at most f(r). If f(r) is a linear or polynomial function, then C has
linear or polynomial expansion. See [41] for background on graph classes with bounded
expansion. Dujmovi´c et al. [21] proved that graphs with bounded layered treewidth have
linear expansion. Indeed, in a graph of layered treewidth k contracting disjoint balls of
radius r gives a graph of layered treewidth at most (4r +1)k, and thus with average
degree O(rk). This result can be extended as follows. Dvoˇr´ak and Norin [23] proved
that for a hereditary graph class G, if every graph G ∈ G has a separator of order
O(|V(G)|1−(cid:15)) for some fixed (cid:15) > 0, then G has polynomial expansion. Lemmas 1 and 11
then imply the following.
Proposition 12. Let G be the class of graphs G such that every subgraph G(cid:48) of G has
k
a path decomposition with magnitude at most k|V(G(cid:48))| and a tree decomposition that
are k-orthogonal. Then G has polynomial expansion.
k
We now show that Proposition 12 cannot be extended to the setting of two tree de-
compositions with bounded intersections. Let G be the 1-subdivision of K , which
n,n
9
has N = n2 + 2n vertices. Say the colour classes of K are V := {v ,...,v } and
n,n 1 n
W := {w ,...,w }. Let x be the division vertex for edge v w . Let S be the star
1 n i,j i j
decomposition of G with root bag {v ,...,v }, and for each j ∈ [n] have a leaf bag
1 n
V ∪{w ,x ,...,x }. Similarly, let T be the star decomposition of G with root bag W,
j 1,j n,j
and for each i ∈ [n] have a leaf bag W ∪{v ,x ,...,x }. Then the intersection of a
i i,1 i,n
bag from S and a bag from T has size at most 3. Each of S and T have magnitude O(N).
On the other hand, contracting the edges incident to each vertex v gives K which
i n,n
has unbounded average degree. Thus the class of 1-subdivisions of balanced complete
bipartite graphs does not have bounded expansion, but every graph in the class has two
tree decompositions with bounded intersections and linear magnitude.
Finally, we consider bounds on pathwidth. A class G of graphs is hereditary if for every
graph G ∈ G every induced subgraph of G is in G. It is well known that hereditary graph
classes with treewidth O(n(cid:15)), for some fixed (cid:15) ∈ (0,1), have pathwidth O(n(cid:15)); see [4]
for example. In particular, Lemma 11 and Lemma 29 of Dujmovi´c et al. [19] imply the
following.
Lemma 13. Let G be a hereditary class of graphs, such that every n-vertex graph G in
G has a tree decomposition T and a weak path decomposition P, such that T and P are
k-orthogonal and P has magnitude at most cn. Then for every n-vertex graph G in G,
√
pw(G) (cid:54) 11 ckn−1.
4 Minor-Closed Classes
Thissectionshowsthatgraphsinafixedminor-closedclasshaveatreedecompositionand
a linear-magnitude path decomposition with bounded intersections. The following graph
minor structure theorem by Robertson and Seymour is at the heart of graph minor theory.
In a tree decomposition T of a graph G, the torso of a bag T is the subgraph obtained
x
from G[T ] by adding, for each edge xy ∈ E(T), all edges vw where v,w ∈ T ∩T . A
x x y
graph G is (g,p,k,a)-almost-embeddable if there is set A of at most a vertices in G
such that G−A can be embedded in a surface of Euler genus g with at most p vortices
of width at most k. (See [21] for the definition of vortex, which will not be used in the
present paper.) A graph is k-almost-embeddable if it is (k,k,k,k)-almost-embeddable. If
G and G are disjoint graphs, where {v ,...,v } and {w ,...,w } are cliques of equal
1 2 1 k 1 k
size respectively in G and G , then a clique-sum of G and G is a graph obtained from
1 2 1 2
G ∪G by identifying v with w for each i ∈ {1,...,k}, and possibly deleting some of
1 2 i i
the edges v v .
i j
10
