SPARSE OBSTRUCTIONS AND EXACT TREEWIDTH DETERMINATION
BRIAN LUCENA
Department of Mathematics
The American University in Cairo
113 Sharia Kasr El Aini
P.O. Box 2511
11511 Cairo, EGYPT
blucena@aucegypt.edu
Abstract. One consequence of the Graph Minor Theorem is that for every k there exists a finite
obstruction set Obs(T W ≤ k). However, relatively little is known about these sets, and very few
general obstructions are known. The ones that are known are the cliques, and graphs which are
“nearly” cliques. This paper gives several general constructions of minimal forbidden minors which
are “sparse” in the sense that the ratio of the treewidth to the number of vertices n does not approach
1 as n approaches infinity. We accomplish this using a novel combination of using brambles to provide
lower bounds and achievable sets to demostrate upper bounds. Additionally, we determine the exact
treewidth of other basic graph constructions which are not minimal forbidden minors.
Keywords: Treewidth, obstruction, graph minor, achievable set, lower bound, upper bound,
bramble, minimal forbidden minor.
1. Introduction
The notion of treewidth is an influential one in contexts both theoretical and applied. On a
theoretical level, treewidth was a fundamental concept in the proof of the celebrated Graph Minor
Theorem of Robertson and Seymour. In more applied realms, treewidth has had an impact on problems
in computer science and statistics, since many graph-based algorithms in those fields are precisely
exponential in the treewidth of the graph in question (meaning they are polynomial on graphs of
bounded treewidth).
A consequence of the Graph Minor Theorem is that for every number k, there exists a finite set of
forbidden minors for treewidth ≤ k. That is, for each k there is a finite set of graphs, Sk , such that
every graph with treewidth > k contains some graph from the set Sk as a minor. The smallest such set,
(i.e. the set that includes only graphs which are minimal in the minor order) is denoted Obs(T W ≤ k)
(or simply Obs(k)) and is referred to as the set of minimal forbidden minors for treewidth ≤ k (or
equivalently the obstruction set for treewidth ≤ k). By definition then, a graph H ∈ Obs(k) if and
only if H has treewidth k + 1 and every minor of H has treewidth k.
These obstruction sets are significant for several reasons. It is known that finding the exact
treewidth of an arbitrary graph is NP-hard [1]. However, since testing minor inclusion has a polynomialtime algorithm (though with an infeasibly large constant), for every k there is a polynomial-time
algorithm to test whether a graph has T W ≤ k. This assumes, however, that the set Obs(k) is known
for that particular k.
Furthermore, the obstruction sets give insight into the structure of graphs of a particular treewidth.
For certain classes of graphs, (e.g. chordal graphs) calculating the exact treewidth is computationally
Date: January 15, 2006.
1
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BRIAN LUCENA
tractable. By knowing and analyzing the sets Obs(k) we could potentially find other such classes of
graphs. As a specific example of this, consider planarity. At this time, it is unknown whether or
not calculating the exact treewidth of a planar graph is NP-hard. There exist planar graphs of every
treewidth, and planarity is closed under taking of minors. Therefore, for every k there exist planar
minimal forbidden minors for treewidth ≤ k. If there are relatively few of them, and the complexity of
testing minor inclusion of these graphs does not grow exponentially with k, then there could potentially
be a polynomial-time (though still practically infeasible) algorithm for computing the exact treewidth
of planar graphs. This is indeed an extremely unlikely scenario, however the point is made that
knowing and understanding these obstruction sets can yield answers to questions of this type. In
general, there has been much recent work on finding lower bounds to treewidth [2] [3] [4] [5] [6] [7] [8] [9].
Knowing the sets Obs(k) could prove invaluable in this regard.
Despite their importance, relatively little is known about the sets Obs(k) or indeed, minimal forbidden minors for treewidth in general. The set Obs(k) is known for k = 1, 2, 3. In the thesis [10] 75
or so minimal forbidden minors for treewidth ≤ 4 are given, and it is believed though not proven that
this may constitute the enitre obstruction set. Furthermore, there are relatively few known general
constructions of minimal forbidden minors to treewidth ≤ k (i.e. constructions that generalize to arbitrarily large k). Trivially, Kn ∈ Obs(n−2). Ramachandramurthi [11] classified the graphs in Obs(k)
which contain exactly k + 3 vertices. Chlebikova [12] generalized the work of Ramachandramurthi,
and gave a clever alternate method to characterize graphs where the number of vertices minus the
treewidth is a fixed constant. However, all these constructions are extremely dense in the sense that
the ratio of the treewidth to the number of vertices approaches 1 as n gets large. In other words, they
are “nearly” cliques.
This paper makes the following contributions: First, we use the theory of brambles to find lower
bounds to graph constructions. We show that in highly symmetric graph constructions, defining a
bramble and proving its order can be an effective way to provide a tight lower bound for treewidth.
Next, we define the notion of an achievable set and show how achievable sets can be used to give upper
bounds for the treewidth of graphs. While specifying achievable sets is essentially no different than
exhibiting a triangulation or a tree-decomposition, we show that in many cases it is much simpler to
demonstrate an achievable set. Combining brambles with achievable sets, we are able to determine the
exact treewidth for several general graph constructions. In most of these cases, we go on to prove that
these constructions are, in fact, minimal forbidden minors for treewidth. The constructions in this
paper are “sparse” in the sense that the treewidth of these constructions is a constant fraction of the
number of vertices. One surprising result is the existence of minimal forbidden minors for treewidth
k with minimum degree 3 for arbitrarily large k.
Section 2 of this paper will give background on treewidth, brambles, introduce the notion of an
achievable set, and review the definition of the Cartesian product of two graphs. In section 3 we
compute the exact treewidth of Kn × Kn . This graph is not a minimal forbidden minor, but in
Section 4, we show that Kn × K3 is, in fact, a minimal forbidden minor for treewidth ≤ 2n − 2.
Furthermore, an entire class of minimal forbidden minors of size quadratic in n can be easily derived
from Kn × K3 . In Section 5, we give a more complicated construction of a minimal forbidden minor
which has minimum degree 3 for arbitrarily large n. Section 6 gives conclusions and ideas for further
work.
2. Preliminaries
All graphs in this paper have no self loops or multiple edges. The neighborhood of a vertex v,
denoted N (v) is the set of all vertices adjacent to v.
For completeness, we will first define a tree decomposition and treewidth.
SPARSE TREEWIDTH OBSTRUCTIONS
3
Definition 2.1. [13] A tree decomposition of G is a family (Xi : i ∈ I) of subsets of VG , together
with a tree T with VT = I with the following properties:
S
(1) i∈I Xi = VG
(2) Every edge of G has both its endpoints in some Xi (i ∈ I)
(3) For i, j, k ∈ I if j lies on the path of T from i to k then Xi ∩ Xk ⊆ Xj
Definition 2.2. [13] The width of the tree-decomposition is maxi∈I |Xi | − 1.
Definition 2.3. [13] The treewidth of G, denoted T W (G) is the minimum k ≥ 0 such that G has a
tree-decomposition of width ≤ k.
Note that there are many equivalent ways to define treewidth. Some of them will be given in the
following sections.
2.1. Brambles. To find lower bounds for treewidth, we use the theory of brambles ( [14] or see [15]
e.g.). Two subsets of vertices of a graph G touch if they contain a vertex in common or if G contains
an edge between them. A bramble is a set of subsets of VG such that every pair touches. A cover C of
a bramble is set of vertices such that every element of the bramble contains a vertex in C. The order
of a bramble B is the minimum cardinality of a cover of B. The bramble number of a graph B(G) is
the largest possible order if a bramble in G. The work in this paper uses heavily the following result
about brambles.
Theorem 2.4. [14] TW(G) = B(G) -1
Specifically, we will use the fact that if B is a bramble of a graph G, and any cover of B has size at
least k, then T W (G) ≥ k − 1.
2.2. Achievable Sets. We introduce a simple method to give an upper bound for the treewidth of a
graph using the notions of border and achievable sets.
Definition 2.5. Given a graph G = (V, E), we define the border of a set of vertices S ⊆ V , denoted
β(S) to be:
(1) If S is connected, then β(S) = |N (S) \ S|.
(2) If S is not connected, then β(S) = maxi β(Ci ) where the Ci are the connected components of
the subgraph of G induced by S.
Definition 2.6. Let π be an ordering of the vertices of G and let vi = π−1 (i). We define the max
border of the ordering π (on the graph G) to be:
(2.1)
M Bπ (G) = max β(Ti )
i
where Ti = {v1 , v2, . . . , vi}
Definition 2.7. Let the minimax border M M B(G) be given by:
(2.2)
M M B(G) =
min
orderings π
M Bπ (G)
Theorem 2.8. MMB(G) = TW(G)
This is fairly well known, and the proof is somewhat tedious, so we omit it. Interested readers can
look at [16], for example.
Definition 2.9. We say a subset S ⊆ V is k-achievable (or achievable at k) if there exists an ordering
π of the vertices in S such that
(2.3)
max β(Ti ) ≤ k
where again Ti = {v1 , v2, . . . , v|S|}.
Lemma 2.10. If S1 and S2 are k-achievable, and S1 and S2 do not touch, then S1 ∪S2 is k-achievable.
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BRIAN LUCENA
The following lemma gives us a simple way to demonstrate that a graph G has T W (G) ≤ k.
Lemma 2.11. A graph G on n vertices has T W (G) ≤ k iff there exists a k-achievable set S of size
n − k − 1.
Proof. If T W (G) ≤ k, find an ordering π with M Bπ (G) = k and let S be the first n − k − 1 vertices
in the ordering. This proves the forward direction. For the backward direction, let π be the ordering
used for S ending with the remaining k + 1 vertices in any order. Clearly M Bπ (G) ≤ k and therefore
T W (G) ≤ k.
The main advantage of examining treewidth from the point of view of borders and achievable sets
is that they are easy to visualize on a graph and do not require altering the graph in any way. It is
straightforward to show that M Bπ (G) is exactly one less than the size of the largest clique formed
by triangulating G using π as an elimination ordering. However, to compute the largest clique in a
triangulation requires drawing in many edges and looking for cliques, whereas computing M Bπ (G)
requires only finding the borders of connected components, which is usually simpler.
As a final bit of background, we give the define the Cartesian product of two graphs. The constructions in this paper are all derived from, if not exactly, Cartesian products of simple graphs.
Definition 2.12. The Cartesian product of two graphs G and H, denoted G × H, is the graph with
vertex set VG×H = VG × VH = {vi,j : i = 1, 2, . . .|VG |; j = 1, 2, . . ., |VH |} and edge set EG×H defined
by {vi1,j1 , vi2 ,j2 } ∈ EG×H if either j1 = j2 and (i1 , i2) ∈ G or i1 = i2 and (j1 , j2) ∈ H.
In other words, for fixed j, the subgraph induced by the vertices {vi,j |i = 1, 2, . . .n} is isomorphic
to G and for fixed i, the subgraph induced by the vertices {vi,j |j = 1, 2, . . .m} is isomorphic to H.
We will sometimes refer to the “copies” of G as columns and the copies of H as rows. Likewise, an
edge contained in a copy of G is called a row edge and an edge contained in a copy of H is called a
column edge.
In the figures used in this paper, vertices contained in an ellipse form a clique, and edges that
extend beyond the vertex are intended to “wrap around”. Furthermore, we use the convention that
the “rows” are numbered from top to bottom and the “columns” from left to right.
3. The Treewidth of Kn × Kn
As a first demonstration of the powerful techniques in this paper, we determine the exact treewidth
of the graph Kn × Kn . For all generally defined graphs in this paper, we assume n ≥ 3.
Theorem 3.1. T W (Kn × Kn ) =
n2
2
+
n
2
− 1 (for n ≥ 3).
2
Proof. We first show that T W (Kn × Kn ) ≤ n2 + n2 − 1. To do this, we prove the following lemma,
which calculates a number k such that an m × m “square” is k-achievable.
Lemma 3.2. Let Qm be the set of vertices defined by {vi,j |1 ≤ i, j ≤ m} in the graph Kn × Kn . Then
the set Qm is (2m(n − m) + m − 1) - achievable for any integer m ≤ n+1
.
2
Proof. We prove this by induction. The base case m = 1 is obvious since the degree of any vertex is
2(n − 1). So assuming the lemma true for 1, 2, . . ., m − 1, we will show Qm is (2m(n − m) + m − 1)achievable. The set Qm−1 is achievable at am−1 = (2(m − 1)(n − (m − 1)) + m − 2). Note that
am−1 ≤ (2m(n − m) + m − 1) for any integer m ≤ n+1
2 . So define an ordering for Qm which
begins with the ordering for Qm−1 followed by (vm,1 , vm,2 , . . . , vm,m−1 , v1,m, v2,m , . . ., vm,m ). The set
Qm−1 ∪ {vm−1 } has a border of size (m(n − m + 1) + (m − 1)(n − m + 1) − 1. The subsequent border
sizes decrease until we reach the subset Qm−1 ∪ {vm,1 , vm,2, . . . , vm,m−1 , v1,m} which has a border of
SPARSE TREEWIDTH OBSTRUCTIONS
5
size 2m(n − m) + m − 1, after which the border decreases until the entire set Qm is achieved. It is
straightforward to verify that:
n+1
.
2m(n − m) + m − 1 ≥ (m(n − m + 1) + (m − 1)(n − m + 1) − 1 for any integer m ≤
2
Therefore Qm is (2m(n − m) + m − 1)-achievable.
If we let Rm = {vi,j |n − m + 1 ≤ i, j ≤ n} it is easy to see, by the symmetry of the graph,
that Rm is also (2m(n − m) + m − 1)-achievable. Thus, if n is even, then Q n2 and R n2 are non2
2
touching, ( n2 + n2 − 1)-achievable sets, and |Q n2 ∪ R n2 | = n2 so by Lemma 2.10 and Lemma 2.11,
2
T W (Kn × Kn ) ≤ n2 + n2 − 1.
n−1
n+1
n2
n
If n is odd, note that a n+1 = 2( n+1
2 )( 2 ) + 2 − 1 = 2 + 2 − 1 and so Q n+1 and R n−1 are
2
2
2
2
2
both ( n2 + n2 − 1)-achievable. Again, by Lemma 2.10 and Lemma 2.11, T W (Kn × Kn) ≤ n2 + n2 − 1.
2
To show T W (Kn × Kn ) ≥ n2 + n2 − 1, we again consider separately the cases where n is odd and
2
even, and exhibit brambles of order at least n2 + n2 .
First, let n be odd. Define a bramble to be every connected subset which contains exactly n+1
2
elements in a single row. This is a bramble since for any two such subsets there exists come column
in which both subsets contain a vertex. Clearly, a cover for such a bramble must contain n+1
vertices
2
in each row for a total of n(n + 1)/2 vertices.
For n even, a more sophisticated bramble is required. First we must construct a set A of subsets
of {1, 2, . . .n} in the following way. For each possible subdivision of {1, 2, . . ., n} into two sets of size
n/2, arbitrarily put one of them (call it A) into A and leave the other (AC ) out of A. Thus A contains
only sets of size n/2 and for any set A ⊆ {1, 2, . . ., n} of size n/2 we have that A ∈ A if and only if
AC ∈
/ A. Note that any two sets in A have a nonempty intersection.
We can now define a bramble B in the following way: Let a connected subset C be in the bramble
B if at least one of the following conditions holds:
(1) C contains vertices in at least n2 + 1 different columns.
(2) C contains vertices in at least n2 + 1 different rows and at least n2 different columns.
(3) C contains vertices in at least n2 different rows, at least n2 columns, and the set of columns
spanned by C is an element of A.
To show this is a bramble, consider any two subsets C1, C2 ∈ B. If either satisfies (1) or (2) above,
then the two subsets must share some row or column in common and therefore touch. If both satisfy
condition (3) only, they must overlap in some column, since any two subsets in A have some element
in common.
2
Now we must show that a cover X of B must have at least n2 + n2 vertices. Note that to cover all
the elements satisfying condition (1) requires a cover to contain at least n2 vertices in each row. So a
2
cover with ≤ n2 + n2 − 1 has at least n2 + 1 rows which contain exactly n2 vertices. There are two cases:
Case 1: There exists some subset of columns S of size n2 such that in every row i, the set of columns
not covered by X, call it Fi, is either a subset of S or a subset of S C . If the number of Fi which are
a subset of S is not n/2, then we have an uncovered connected component with vertices in at least
n
n
2 + 1 different rows and at least 2 different columns, meaning a component satosfyong (2) above is
n
uncovered. Otherwise, in exactly 2 rows, the cover X contains vertices in each of the columns in the
set S and in the other n2 columns, the cover contains vertices in each of the columns in the set S C .
2
So n2 vertices are used to cover S or S C in the respective rows, leaving only n2 − 1 other vertices in
this cover. Since either S or S C is in A and n2 − 1 additional covered vertices cannot reduce the span
(in rows or columns) of the n2 × n2 component, there exists a uncovered component meeting condition
(3) above.
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BRIAN LUCENA
Case 2: Case 1 does not happen. There are at least n2 + 1 rows which have exactly n2 uncovered
vertices. Call this set of rows R. We will argue that the uncovered vertices in these rows are all
connected (in G − X). If not, then there exists a set S of size n2 such that some of those rows have
set S covered and the rest have set S C . If there are 3 different sets covered in the rows R, or two
non-complementary sets, then all uncovered vertices in those rows will be connected. Moreover, if any
row (including a row in RC ) has an uncovered set which is not a subset of either S or S C then all
the uncovered vertices in the rows in R are connected, as well. Therefore all uncovered vertices in the
rows R will be connected unless we are in case 1 above. The uncovered vertices in the rows in R form
a component meeting condition (2) above, and therefore some set in the bramble is uncovered.
This concludes the proof.
Theorem 3.3. Kn × K3 is a minimal forbidden minor for graphs of treewidth less than 2n − 1 (for
n ≥ 3).
Figure 1. The graph Kn × K3 .
Proof. We have to show that T W (Kn × K3 ) = 2n − 1 and that any graph G0 formed by removing or
contracting any edge from Kn × K3 yields a graph with T W (G0) ≤ 2n − 2.
To show that T W (Kn ×K3 ) ≤ 2n−1, we verify that the set S = {v1,1, v2,1, . . . , vb n2 c,1 , vb n2 c+1,2 , vb n2 c+2,2 . . . , vn,2}
is (2n−1)-achievable. Since |S| = n and |VKn ×K3 | = 3n, by Lemma 2.11 we have that T W (Kn ×K3 ) ≤
2n − 1.
To show T W (Kn × K3 ) ≥ 2n − 1, we will construct a bramble B such that the order of B is at
least 2n. Let B contain every subset of size 2 which contains vertices in two different “columns”. In
other words, {vi1 ,j1 , vi2 ,j2 } ∈ B iff i1 = i2 and j1 6= j2. Since there are only three columns, for any
two subsets in B, there exists a column such that both subsets contain some vertex in that column.
Therefore, any two subsets in B touch and thus B is a bramble.
To demonstrate that the order of B is at least 2n, simply note that any set that covers B must
contain two vertices in each row. Thus T W (Kn × K3) ≥ 2n − 1.
To show minimality we show that in each of the following four cases, the resulting graph has
treewidth ≤ 2n − 2:
(1) A row edge is removed. Without loss of generality, assume the edge is (v1,1, v1,2). Let S1 =
{v1,1}, S2 = {v1,2, v2,2, . . . , vb n2 c,2 } and S3 = {vb n2 c+1,3, vb n2 c+2,3 , vn,3}. It is straightforward
to verify that S1 , S2 and S3 are 2n − 2-achievable and do not touch one another. Therefore
the set S = S1 ∪ S2 ∪ S3 is 2n − 2-achievable and of size n + 1. Use Lemma 2.11 to show the
treewidth is ≤ 2n − 2.
(2) A row edge is contracted. Without loss of generality, assume the edge is (v1,2, v1,3). Let
S1 = {v1,1, v2,1, . . ., vb n2 c,1 } and S2 = {vb n2 +1c,2 , vb n2 c+2,2, vn,2}. Then S = S1 ∪ S2 is 2n − 2achievable and its size is n. Since the resulting graph has only 3n − 1 vertices, we can use
Lemma 2.11 to show the treewidth is ≤ 2n − 2.
SPARSE TREEWIDTH OBSTRUCTIONS
7
(3) A column edge is removed. Without loss of generality, assume the edge is (vb n2 c,2, vb n2 c+1,2 ).
Let S1 = {v1,1, . . . , vb n2 c,1, vb n2 c,2 } and S2 = {vb n2 c+1,3 , . . ., vn,3}. Note that β(S1 ) contains
n − 2 vertices in column 2 and n total vertices in columns 1 and 3, therefore S1 is 2n − 2achievable as is S3 . Since S1 and S2 do not touch, S = S1 ∪ S2 is 2n − 2-achievable and
|S| = n + 1. Use Lemma 2.11.
(4) A column edge is contracted. Without loss of generality, assume the edge is (v1,1, v1,2). Let
S1 = (v1,3, v1,4, . . . , v1,n}, and S2 = {v2,1, v2,2}. Then S1 and S2 are 2n − 2-achievable and do
not touch, so the set S = S1 ∪ S2 is 2n − 2-achievable and of size n. Use Lemma 2.11.
Thus the theorem follows.
Figure 2. An example of a graph in D5 .
Theorem 3.4. Let Dn be the class of graphs that can be formed by adding a vertex v∗ to Kn × K3 and
adding edges between v∗ and exactly 2 vertices in each row. Let Dn ∈ Dn . Then Dn ∈ Obs(T W ≤
2n − 1).
Proof. To show T W (Dn ) ≥ 2n, let B be the bramble from the previous proof (Theorem 3.3) and let
0
B = B ∪ {v∗ }. Since v∗ is adjacent to two vertices in each row, v∗ touches every set in B, therefore
0
0
B is a bramble. Since v∗ must be covered, and it takes 2n vertices to cover B, the order of B is
2n + 1. To show T W (Dn ) ≤ 2n, let S = VDn − {v∗ ∪ N (v∗ )}. It is straightforward to verify that S is
2n-achievable and |S| = n and so Lemma 2.11 gives us T W (Dn ) ≤ 2n.
To show minimality, first note that any subset of vertices in Kn × K3 which is k-achievable in
Kn × K3 is (k + 1)-achievable in Dn , since there is at most one additional vertex that any set could
touch. Therefore, removing or contracting any edge not having v∗ as an endpoint will reduce the
treewidth using the same arguments as in the proof of Theorem 3.3. Thus we only need consider the
cases where we remove or contract an edge with v∗ as an endpoint. If we contract such an edge, the
set S = VDn − {v∗ ∪ N (v∗ )} is now (2n − 1)-achievable (since v∗ and its neighbor now only “count” as
one vertex. Since the resulting graph also now has one fewer vertex, Lemma 2.11 shows the resulting
0
treewidth is ≤ 2n − 1 If we remove such an edge to make a graph Dn , it is straightforward to verify
0
that the set of vertices S = VDn0 − {v∗ ∪ N (v∗ ) is (2n − 1)-achievable (just end the ordering with
0
the two vertices that are in the same row, in the appropriate order). Note that |S | = n + 1 and use
Lemma 2.11.
A few moments thought reveals that |Dn| equals the number of integer partitions of n of size less
than or equal to three. This is because for each row, there are 3 possibilities for which two vertices
are covered, yet neither the order of the rows nor the “naming” of the three possiblities should be
considered due to the symmetry of the construction. Without going into details, the number of ways
to partition n into 3 or less numbers is quadratic in n, and therefore so is the size of Dn.
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BRIAN LUCENA
4. A Minimal Forbidden Minor with Minimum Degree 3
In the following, we give a slightly more complicated construction to construct another minimal
obstruction for treewidth less than or equal to 2n − 1. We will motivate this construction by first
presenting and proving properties about some simpler graphs.
4.1. The Treewidth of Kn × C4.
Proposition 4.1. Let Cn be a simple cycle on 4 vertices. The graph Kn × C4 has treewidth 2n − 1.
Proof. To show that the treewidth is less than or equal to 2n − 1, Let S1 = {v1,1, . . . , vb n2 c , 1}, S2 =
{vb n2 c+1,2 , . . . , vn,2}, S3 = {v1,3, . . . , vb n2 c,3 }, and S4 = {vb n2 c+1,4 ), . . . , vn,4}. Each of S1 , S2, S3 , and S4
are (2n − 1)-achievable and no pair touches one another. Therefore S = S1 ∪ S2 ∪ S3 ∪ S4 is
(2n − 1)-achievable, and furthermore |S| = 2n. Since |VKn ×C4 | = 4n we can use Lemma 2.11 to
show T W (Kn × C4) ≤ 2n − 1.
To show that the treewidth is greater than or equal to 2n − 1, we will give a bramble B of order
2n. Let B contain the following subsets of vertices:
(1) {vi,1, vi,2, vi,3} for i = 1, . . ., n.
(2) {vi,2, vi,3, vi,4} for i = 1, . . ., n.
(3) {vi,3, vi,4, vi,1} for i = 1, . . ., n.
(4) {vi,4, vi,1, vi,2} for i = 1, . . ., n.
Clearly this is a bramble, since every subset contains vertices in three different columns and so
for any two subsets there exists some column (actually two columnss) such that both subsets contain
vertices in that column and therefore touch. Now consider any cover of the bramble. Clearly it must
contain at least two vertices in each row for a total of at least 2n vertices.
4.2. Augmenting Kn ×C4 . Let the graph An be given by adding the edges (v1,1 , v1,3) and (v1,2, v1,4)
to Kn × C4 . We will refer to these two edges as the “additional” edges.
Figure 3. The graph A5 .
Theorem 4.2. The graph An given above has treewidth 2n.
Proof. Let S1 = {v2,2, v3,2, . . . , vn,2}, S2 = {v2,4, v3,4, . . . , vn,4} and S3 = {v1,1, v1,3}. Each of S1 , S2,
and S3 is 2n-achievable and they are mutually non-touching. Therefore S = S1 ∪ S2 ∪ S3 is 2nachievable and has size 2n, so we can use Lemma 2.11 to show T W (An ) ≤ 2n.
To show T W (An ) ≥ 2n we will give a bramble of order 2n + 1. Let B be the bramble defined in
the proof of Proposition 4.1 and let B0 = B ∪ {{v1,1, v1,2}, {v1,2, v1,3}, {v1,3, v1,4}, {v1,4, v1,1}}. It is
straightforward to verify that this is still a bramble. Note that any bramble cover must contain at
least 2 vertices in each of rows 2 through n and 3 vertices in row 1 for a total of 2n + 1 vertices. Proposition 4.3. Contracting any edge in An yields a graph A0n with treewidth ≤ 2n − 1.
SPARSE TREEWIDTH OBSTRUCTIONS
9
Proof. By the symmetry of the graph, we only have to consider five cases of edges to contract.
(1) We contract a “row” edge in rows 2 through n. Assume WLOG that the edge is (v2,1, v2,2).
Let S1 = {v2,3, v3,3, . . . vn,3, v2,4, v3,4, . . . , vn,4} and let S2 = {v1,1}. S1 is (2n − 1)-achievable
(e.g. by the ordering (v2,3 , v2,4, v3,3, v3,4, . . ., vn,3, vn,4) and |S1| = 2n − 2, while S2 is also
(2n − 1)-achievable, has size 1 and does not touch S1 . Therefore S = S1 ∪ S2 is (2n − 1)achievable and of size 2n − 1. Since A0n has 4n − 1 vertices, by Lemma 2.11 we have that
T W (A0n ) ≤ 2n − 1.
(2) We contract a “row” edge in row 1. Assume WLOG that the edge is (v1,1, v1,2). Let S1 =
{v2,1, v3,1, . . . vn,1, v2,2, v3,2, . . . , vn,2} and S2 = {v1,3}. S = S1 ∪ S2 is (2n − 1)-achievable and
|S| = 2n − 1. Use Lemma 2.11.
(3) We contract a “column” edge involving a row 1 vertex. Assume WLOG the edge is (v1,1, v2,1).
Then S = {v1,3, v2,3, . . . , vn,3, v1,4, v2,4, . . . , vn−1,4} is (2n − 1)-achievable (e.g. use an ordering
that begins v1,3, v2,4, v1,4, . . .) and |S| = 2n − 1. Use Lemma 2.11.
(4) We contract a “column” edge not involving a row 1 vertex. Assume WLOG the edge is
(v2,1 , v3,1). Let S1 = {v2,2, v3,2, . . . , vn,2, v2,3, v3,3, . . . , vn,3} and S2 = {v1,4}. S1 is (2n − 1)achievable as is S2 and the two sets do not touch. Let S = S1 ∪ S2 and use Lemma 2.11.
(5) We contract one of the two “additional” edges. Assume WLOG the edge is (v1,1, v1,3). Let
S1 = {v2,1, v3,1, . . . , vn,1}, S2 = {v2,3, v3,3, . . . , vn,3} and S3 = {v1,2, v1,4}. All three sets S1 ,S2,
and S3 are now (2n − 1)-achievable (recall that v1,1 and v1,3 together now “count” as only
one vertex). Let S = S1 ∪ S2 ∪ S3 , |S| = 2n so we can use Lemma 2.11.
Proposition 4.4. Removing any row edge in An yields a graph A0n with treewidth ≤ 2n − 1.
Proof. There are two cases to consider:
(1) We remove a “row” edge in row 1. Assume WLOG the edge is (v1,1, v1,2). Let S1 = {v1,1, v2,1},
S2 = {v1,2, v3,2, v4,2, . . . , vn,2}, S3 = {v2,3}, and S4 = {v3,4, v4,4, . . . , vn,4}. Each of these sets
is (2n − 1) achievable and no pair touches one another. Let S = S1 ∪ S2 ∪ S3 ∪ S4 , then
|S| = 2n, S is (2n − 1)-achievable, and by Lemma 2.11 T W (A0n ) ≤ 2n − 1.
(2) We remove a “row” edge in rows 2 through n. Assume WLOG the edge is (v2,1, v2,2). Let S1 =
{v1,1, v2,1}, S2 = {v2,2, v3,2, . . . , vn,2, v2,3}, and S3 = {v3,4, v4,4, . . . , vn,4}. It is straightforward
to verify that each set is (2n − 1)-achievable and no two touch. Letting S = S1 ∪ S2 ∪ S3
we have that S is (2n − 1)-achievable and |S| = 2n. Therefore, by Lemma 2.11 we have
T W (A0n ) ≤ 2n − 1.
Proposition 4.5. Removing either “additional” edge in An yields a graph A0n with treewidth ≤ 2n−1.
Proof. Assume WLOG the edge is (v1,1 , v1,3). Use the same achievable sets as in the proof of Proposition 4.1 (defined on Kn × C4).
Proposition 4.6. Removing a column edge involving a row 1 vertex from An yields a graph A0n with
treewidth ≤ 2n − 1.
Proof. Assume WLOG that the edge is (v1,1, v2,1). Let S1 = {v1,1, v1,4, v3,4, v4,4 . . . , vn,4} and S2 =
{v2,1, v2,2, v3,2, . . . , vn,2}. The two sets do not touch and are each (2n − 1)-achievable, so S = S1 ∪ S2
is a (2n − 1)-achievable set and its size is 2n. Use Lemma 2.11.
As it turns out, removing a single column edge not involving a row 1 vertex does not yield a graph
with treewidth ≤ 2n − 1. This is easily seen since the bramble B0 from the proof of Theorem 4.2
remains a bramble if such an edge is removed. Therefore, An is not itself a minimal forbidden minor.
10
BRIAN LUCENA
However, removing some sequence of column edges (not involving row 1 vertices) will eventually lead
to a minimal forbidden minor. In fact the graph An contains at least several different such minimal
forbidden minors. Here we exhibit one such graph.
Theorem 4.7. Let Qn be the graph formed by removing every edge of the form (vi,1 , vj,1) for 2 ≤ i 6=
j ≤ n. Then Qn is a minimal forbidden minor for treewidth ≤ 2n − 1.
Figure 4. The graph Q5 .
Proof. Clearly, T W (Qn ) ≤ 2n since Qn is a subgraph of An . The bramble B0 from the proof of
Theorem 4.2 remains a bramble for Qn, which shows that T W (Qn) ≥ 2n. To show minimality, by
Propositions 4.4, 4.5, 4.6, and the fact that Qn is a subgraph of An , we need only consider the case
where we remove a “column” edge not involving row 1. Since, by the construction of Qn , all column
edges in column 1 use a row 1 vertex, and since column 2 is symmetrical to column 4, we are left with
two subcases:
(1) An edge of the form (vi,3, vj,3) is removed for some 2 ≤ i 6= j ≤ n. WLOG assume the
edge is (v2,3, v3,3). Let S1 = {v2,3, v2,4, v2,1}, and consider the ordering (v2,3, v2,1, v2,4). The
border of S1 contains 1 vertex in column 1, (v1,1 ), 1 vertex in column 2 (v2,2 ), n − 2 vertices
in column 3 (every vertex except v2,3 and v3,3) and n − 1 vertices in column 4 for a total of
2n − 1. So S1 is (2n − 1)-achievable. Let S2 = {v3,1, v4,1, . . . , vn,1, v3,2, v4,2, . . . , vn,2, v3,3}.
S2 is (2n − 1)-achievable by the ordering (v3,3, vn,2, vn−1,2, . . . v3,2, v3,1, v4,1, . . . , vn,1). S1 and
S2 do not touch and |S1 ∪ S2 | = 3 + 2n − 3 = 2n. So S = S1 ∪ S2 and by Lemma 2.11 the
resulting graph has treewidth ≤ 2n − 1.
(2) An edge of the form (vi,2, vj,2) is removed for some 2 ≤ i 6= j ≤ n. WLOG assume the
edge is (v2,2 , v3,2). Let S1 = {v2,1, v2,2, v2,3}, then S1 is (2n − 1)-achievable. Let S2 =
{v3,2, v3,1, v4,1, . . . , vn,1, v3,4, v4,4, . . . , vn,4}. S2 is also (2n − 1)-achievable by the ordering
(v3,4 , v4,4, . . . , vn,4, vn,1, vn−1,1, . . . , v4,1, v3,2, v3,1). Again, since S1 and S2 do not touch, S =
S1 ∪ S2 is (2n − 1)-achievable and |S| = 2n. Use Lemma 2.11.
Therefore, any edge contraction or removal lowers the treewidth of Qn.
Corollary 4.8. For arbitrarily large k, there exists G ∈ Obs(k) with minimum degree 3.
It is an easy consequence of well-known results that a graph in Obs(T W (k)) cannot have a vertex
of degree 1 or 2 for k ≥ 3.
5. Conclusions and Further Work
We show that using brambles to establish lower bounds for treewidth and achievable sets to give upper bounds is an effective method for determining the exact treewidth of general graph constructions.
Using this approach, we are able to prove the exact treewidth of Kn × Kn , Kn × K3 , and Kn × C4
and a variety of related graphs. We are also able to show that a variety of graph constructions are
SPARSE TREEWIDTH OBSTRUCTIONS
11
actually minimal forbidden minors for treewidth. We show that Kn × K3 ∈ Obs(T W ≤ 2n − 2) and
furthermore define an entire class of minimal forbidden minors for treewidth ≤ 2n − 1 which has a
size quadratic in n. Finally, we exhibit a more complicated construction, that in addition to being
a minimal forbidden minor for treewidth ≤ 2n − 1 has minimum degree. This establishes that for
arbitrarily large k there exists graphs G ∈ Obs(T W ≤ k) which have minimum degree 3, a somewhat
unexpected result.
Ideas for further work are related to learning and understanding the set Obs(T W ≤ k) for all k.
This paper suggests that although these sets grow rapidly with k, the graphs may fall into easily
understandable classes. Therefore, this goal may not be quite so daunting. Moreover, there are many
interesting properties of minimal forbidden minors that are yet to be understood, such as degree
properties, diameter, and girth. Finally, general techniques for constructing further classes of these
graphs need to be developed.
6. Acknowledgements
Some of the work for this paper was done while the author was at UC-Berkeley on an NSF Mathematical Sciences Postdoctoral Research Fellowship.
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12
BRIAN LUCENA
The American University in Cairo, Department of Mathematics, 113 Sharia Kasr El Aini, P.O. Box 2511,
11511 Cairo, EGYPT
E-mail address: blucena@aucegypt.edu