In this section, some preliminaries concerning tree decompositions are presented.
Definition 2.11.
LetG= (V, E) be a graph. A pair (T,X) withX = (Xi)i∈V(T)is atree decompositionofGifT is a tree, Xi⊆V for every i∈V(T), and the following three properties are satisfied.
(T1) For everyv∈V, there is somei∈V(T) withv∈Xi. (T2) For everye∈E, there is somei∈V(T) withe⊆Xi.
(T3) For alli, j, h∈V(T), ifhis on the (unique)i,j-path inT, thenXi∩Xj⊆Xh. Thewidthof a tree decomposition (T,X) is max
|Xi| −1: i∈V(T) . Thetree-widthtw(G) of a graphG is the smallest integertsuch thatGallows a tree decomposition of widtht.
Consider a tree decomposition (T,X) with X = (Xi)i∈V(T) of a graph G. To easily distinguish the vertices ofGfrom the vertices ofT, the vertices in V(T) are callednodes in the following. Furthermore, fori∈V(T), the setXiis referred to as thecluster ofiin (T,X). It is easy to show that (T3) is equivalent to the following condition, see also Section 2 in [Bod98]:
(T3’) For everyv∈V, the graph T[Iv] withIv={i∈V(T): v∈Xi} is connected.
In the following, (T1), (T2), (T3), and (T3’) always refer to the properties defined here.
Every graphG= (V, E) has a tree decomposition. For example, letT = ({i},∅) andXi=V, then (T,X) where X consists only ofXiis a tree decomposition ofG. It follows that tw(G)≤n−1 for every graphG onn vertices. If a graphGallows a tree decomposition of width 0, thenE(G) =∅. Next, it is shown that every tree ˜T with at least two vertices has tree-width 1. Let ˜T = ( ˜V ,E) be a tree with˜ |V˜| ≥2. To construct a tree decomposition of ˜T, let T be the tree obtained from ˜T by subdividing each edge of ˜T once. For each e∈E, denote by˜ ie the vertex used to subdivide e. For eachv ∈V˜, defineXv ={v}
and, for each e={v, w} ∈ E, define˜ Xie ={v, w}. Let X = (Xi)i∈V(T). Clearly, (T,X) satisfies (T1) and (T2). To see that (T3’) is satisfied, letv∈V˜, thenv∈Xi if and only ifi=vori=ie for an edgee that is incident tovin ˜T. Hence, for each v∈V˜, the set{j∈V(T): v∈Xj}={v} ∪ {ie: e∈E, v˜ ∈e}
induces a connected subgraph ofT, namely a star or an isolated vertex. Consequently, (T,X) is a tree decomposition of ˜T and tw( ˜T)≤1. As ˜T contains at least one edge, tw( ˜T) must be at least 1 due to (T2).
Next, a tree decomposition of the square grid is presented. Fix an integerk≥2 and let ˜G= ( ˜V ,E) be˜ thek×k grid. Recall that ˜V =(˜i,˜j): ˜i∈[k],˜j∈[k] . For eachi∈[k−1] and eachj∈[k] define
X(i−1)k+j :=(i,˜j)∈V˜: ˜j≥j ∪(i+ 1,˜j)∈V˜: ˜j≤j ,
see Figure2.3for a visualization. LetT be the path obtained fromPk(k−1) by removing the node 0 and letX = (Xh)h∈[k(k−1)]. For all ˜i,˜j∈[k],
(˜i,˜j)∈Xh ⇔ h∈(˜i−2)k+ ˜j, . . . ,(˜i−1)k+ ˜j ∩[k(k−1)].
It follows that (T1) and (T3’) are satisfied. To see that (T2) is satisfied, lete∈E. If˜ e={(˜i,˜j),(˜i+ 1,˜j)}
for some ˜i ∈[k−1] and ˜j ∈ [k], then e ⊆Xh for h= (˜i−1)k+ ˜j. Otherwise,e ={(˜i,˜j),(˜i,˜j+ 1)} for some ˜i∈[k] and ˜j ∈[k−1] and e⊆Xh for h= (˜i−1)k+ ˜j. Consequently, (T,X) satisfies (T2) and (T,X) is a tree decomposition of ˜Gof widthk. Hence, tw( ˜G)≤k. A matching lower bound is stated in Lemma 88 in [Bod98] or Exercise 21 in Chapter 12 of [Die12]. This yields the next proposition.
Proposition 2.12.
a) Each tree T on at least2 vertices satisfiestw(T) = 1.
(1,1)
(5,1) (1,5)
(5,5) X1
1 = (1−1)5 + 1
X2 2 = (1−1)5 + 2
X8
8 = (2−1)5 + 3
X16
16 = (4−1)5 + 1 X20
20 = (4−1)5 + 5 j
i
Figure 2.3:A few clusters of a tree decomposition of the 5×5 grid of width 5.
A tree decomposition (T,X) of a graph G is a path decomposition of G if T is a path. The tree decomposition of the square grid that was presented above is a path decomposition. The width of a path decomposition is defined as the width of a tree decomposition. Thepath-width of a graphGis the smallest integertsuch thatGallows a path decomposition of widtht and is denoted by pw(G). Clearly, all graphsGsatisfy pw(G)≥tw(G).
For planar graphs, the following bound on the tree-width is known. Its proof can be found in [Bod98].
There, Corollary 23 states that the path-width of a planar graph onnvertices is at mostO(√
n). Using that tw(G)≤pw(G) for all graphsGimplies the next proposition.
Proposition 2.13.
Every planar graphGon nvertices satisfiestw(G) =O(√ n).
Consider a graphG= (V, E), a tree decomposition (T,X) withX = (Xi)i∈V(T), and a graphH ⊆G. A tree decomposition forHcan be easily obtained from (T,X) by deleting all vertices that are not inH. More precisely, fori∈V(T), let ˜Xi=Xi∩V(H). Then, it is easy to check that (T,X˜) with ˜X = ( ˜Xi)i∈V(T)
is a tree decomposition ofH. The tree decomposition (T,X˜) is called theinduced tree decomposition ofH with respect to (T,X). Observing that the width of (T,X˜) is at most the width of (T,X) yields the next proposition.
Proposition 2.14.
LetGbe a graph and let(T,X)be a tree decomposition ofGof widtht−1. For every subgraphH⊆G, the induced tree decomposition ofH with respect to(T,X)is a tree decomposition of H of width at mostt−1.
Furthermore,tw(H)≤tw(G).
Similarly to the construction for subgraphs, a tree decomposition of a minor can be constructed, see also Lemma 12.3.3 and Proposition 12.3.6 in [Die12]. The following proposition is obtained.
Proposition 2.15.
For all graphsH andG such thatGcontainsH as a minor, tw(H)≤tw(G).
Often, when constructing a cut in a tree ˜T, the vertex set of ˜T is partitioned by removing all edges incident to a vertexv∈V( ˜T) and considering the vertex sets of the resulting components. Then, a cut in ˜T of width at most ∆( ˜T) is obtained when combining these vertex sets in an arbitrary way. This idea can be generalized by considering clusters of a tree decomposition, as done in the next lemma. It uses the following notation: Consider a graphG= (V, E) and a tree decomposition (T,X) ofG. For each nodei inT define
EG(i) =
e∈E:e∩Xi6=∅ and eG(i) =|EG(i)|,
where Xi denotes the cluster of i in (T,X). Observe that, when t−1 denotes the width of (T,X), theneG(i)≤ |Xi|∆(G)≤t∆(G) for everyi∈V(T). We say that two subgraphsH1⊆GandH2⊆G are disjoint parts of G if V(H1)∩V(H2) = ∅ and there is no edge e = {v, w} in G withv ∈ V(H1) andw∈V(H2). Note that, ifGis not connected, then two distinct components ofGare disjoint parts ofG, but the subgraphHi fori∈ {1,2}in the definition of disjoint parts does not have to be connected.
The next lemma says that, if the vertices inXior the edges inEG(i) are removed for somei∈V(T), then the graph Gsplits into several disjoint parts. So, these disjoint parts can be combined in an arbitrary way to obtain a cut inGof width at mosteG(i)≤t∆(G). The lemma is a widely known fact about tree decompositions, similar statements are Fact 10.13 and Fact 10.14 in [KT06] or Corollary 1.8 in [Ree97].
Lemma 2.16.
LetG= (V, E)be an arbitrary graph and let(T,X)be a tree decomposition ofGwithX = (Xj)j∈V(T). Fix an arbitrary nodei∈V(T), letk:= degT(i), and denote byi1, i2, . . . , ik the neighbors ofiinT. For`∈[k], letV`T be the node set of the component ofT−ithat contains i` and define V` :=S
j∈V`TXj\Xi. a) Removing the vertices inXi from GdecomposesGintok disjoint parts, which areG[V1], . . . , G[Vk].
b) Removing the edges inEG(i)fromGdecomposesGintok+|Xi|disjoint parts, which are ({v},∅) for every v∈Xi andG[V`] for every`∈[k].
Proof. LetG= (V, E), (T,X) withX = (Xj)j∈V(T),i,k,i1, . . . , ik, as well asV`T andV`for each`∈[k]
be as in the statement.
a) For eachv∈V letIv:={j∈V(T): v∈Xj}, which is the same as in (T3’). Due to (T1), it follows that
[
`∈[k]
V` =
[
j∈V(T)\{i}
Xj
\Xi = V \Xi.
Consider a vertex v∈V. Ifv∈Xi, thenv /∈V` for every`∈[k]. Otherwise,v /∈Xi andIv⊆V`T for a unique `∈[k] asT[Iv] is connected by (T3’), nonempty by (T1), and does not contain the node i. So v ∈V` impliesv /∈ V`0 for all `0 6= ` and therefore the setsV1, . . . , Vk are a partition ofV \Xi.
It remains to show that, for all distinct `1, `2 ∈[k], there is no edge{v1, v2} ∈E withv1 ∈V`1
andv2∈V`2. Assume, for a contradiction, that there is such an edge inE. Property (T2) says that there is a nodej∗ withv1∈Xj∗ andv2∈Xj∗. So,j∗∈Iv1∩Iv2. Alsoj∗6=ibecausev1∈/Xi. As argued before,Iv1 ⊆V`T1 andIv2 ⊆V`T2, because v1∈/Xi andv2∈/ Xi. As`16=`2, the treesT[V`T1] andT[V`T2] are different components ofT−i. Hence,V`T1 ∩V`T2 =∅ and thereforeIv1∩Iv2=∅, but this contradicts that j∗∈Iv1∩Iv2.
b) Clearly, every vertexv∈Xi is an isolated vertex inG−EG(i). So, it suffices to show thatG−Xi decomposes into the kdisjoint partsG[V1], G[V2], . . . , G[Vk], which is equivalent to Parta). 2
The next proposition says that low tree-width implies that a graph does not have many edges.
Proposition 2.17 (Fact 1.10 in [Ree97]).
Every graphG onnvertices satisfies|E(G)| ≤ntw(G). For a proof see Fact 1.10 in [Ree97].