Discrete Mathematics and Theoretical Computer Science 3, 1999, 141–150
A note on domino treewidth
Hans L. Bodlaender
Department of Computer Science, University of Utrecht, P.O.Box 80.089, 3508 TB Utrecht, the Netherlands. Email:
hansb@cs.uu.nl.
received August 11, 1998, revised May 14, 1999, accepted May 27, 1999.
In [DO95], Ding and Oporowski proved that for every , and , there exists a constant , such that every graph
with treewidth at most and maximum degree at most has domino treewidth at most . This note gives a new
simple proof of this fact, with a better bound for , namely .
It is also shown that a lower bound of holds: there are graphs with domino treewidth at least !"# ,
treewidth at most , and maximum degree at most , for many values and . The domino treewidth of a tree is at
most its maximum degree.
Keywords: treewidth, domino treewidth, graph algorithms, tree decompositions
1 Introduction
In [DO95], Ding and Oporowski proved that for every $ and % , every graph &
'
( )+*-,". with
treewidth at most $ and with maximum degree at most % has a tree decomposition of width at most
/10!2
(435656$78%:96*<;>=:556%69. , such that every vertex ?A@B) belongs to at most two of the sets associated to
the nodes in the tree decomposition. Such a tree decomposition was called a domino tree decomposition
by Bodlaender and Engelfriet in [BE97], where they independently gave a similar result, but with a more
complicated proof and with a much higher constant, which was exponential, both in $ and in % .
In this note, a new and easy to understand proof for the result is given. Additionally, the constant
factor arising from the proof given here is smaller: it is shown that graphs with treewidth at most $ and
maximum degree at most % have domino treewidth at most ( C6$ED#F!.G%H( %IDKJL.NMOJ .
The proof uses amongst others a technique from [BGHK95] (inspired by a technique from [RS95]),
and some other ideas. The proof is given in Section 3.
In Section 4, it is shown that a lower bound of PQ( $%R. holds: there are graphs with domino treewidth at
least S $T%UMOJ , treewidth at most $ , and maximum degree at most % , for many values $ and % .
S 7
Some
final remarks are made in Section 5, and it is shown that the domino treewidth of a tree is at most
its maximum degree.
V
This research was partially supported by ESPRIT Long Term Research Project 20244 (project ALCOM IT: Algorithms and
Complexity in Information Technology).
W
1365–8050 c 1999 Maison de l’Informatique et des Mathématiques Discrètes (MIMD), Paris, France
142
Hans L. Bodlaender
2 Definitions and Preliminary Results
Definition. A tree decomposition of a graph &X'X()Y*<,Z. is a pair (\[^]`_baTcd@feTg:*Ghi'X( ej*<kU.G. with
[]`_lacY@mejg a collection of subsets of ) , and hn'o( ej*<kU. a tree, such that
pAq
p
_4rs ]
_ 't)
p
for all edges (4?H*-uI.@m,
there is an c+@ve with ?w*Gux@v] _
for all c*zy*$d@{e : if y is on the path from c to $ in h , then ] _j| ]`}Z~O]Z .
The width of a tree decomposition (\[]`_€a!cU@AeTg:*Gh'i(4eH*-kU.G. is /`0>2 _4rs‚a ]1_<a:MtJ . The treewidth of a
graph &x'ƒ( )+*-,Z. is the minimum width over all tree decompositions of & .
In some cases, h will be considered a rooted tree; a specific node of h is considered to be the root. A
tree decomposition ( „v*-hI. with h a rooted tree is called a rooted tree decomposition. For a node cI@e ,
we call the set ] _ the bag of c .
Definition. A tree decomposition (\[^]`_YaLc…@ejg6*-hB'†(4eH*-kU.-. of &†'†( )+*-,Z. is a domino tree decomposition, if for each vertex ?d@) , there are at most two nodes c‚@‡e with ?{@]`_ . The domino treewidth
of a graph &o'f()Y*<,Z. is the minimum width over all domino tree decompositions of & .
The open neighbourhood of a set of vertices ˆ‰~O)
Š
in a graph &f'o()Y*<,Z. is
(ˆf.+'B[^?‹@)BMŒˆa^ŽRuf@ˆ‰[?w*GuEgU@v,1g
For a graph &o'f()Y*<,Z. , and ˆ~K) , the subgraph of & , induced by ˆ
is denoted as
&` ˆ’‘'f(ˆŒ*[6[?w*GuEgU@v,“a?w*Guf@mˆ“gL.
Lemma 2.1 Let hK'ƒ(4eH*-kU. be a tree. Let ”
p
p
a•”
p
”
S
7
a–#—™˜6a•”
~’”
7
S
a>MOJ
~Ae
S
. Then there exists a set ”
7
~#e
with
.
.
Every subtree of h" eEMA” ‘ is adjacent to at most two nodes in ” .
7
7
Proof: Choose an arbitrary root š in h .
Let ” 'x”
[y`@ve`a-y is the lowest common ancestor of two nodes in ” g . We claim that this set ”
7
7
Sl›
S
fulfils the conditions. Clearly, ” ~t” .
7
S
Let hIœ be a subtree of hZ eQM” ‘ . If czžE@” is adjacent to a node in hbœ , then there are two cases:
7
p
c ž
p
7
is an ancestor of a node in hbœ . Then c ž is the unique parent of the root of hIœ .
c ž is a child of a node in hbœ . We claim that there can be only one node fulfilling this case (for this
tree hbœ ): suppose c ž @” and c @“” are children of nodes in hbœ . Then, the lowest common
7
7
S
ancestor c of c ž and c belongs to hbœ . However, c ž and c belong to ” or are ancestor of a node
7
S
S
S
in ” . So, c is the lowest common ancestor of two nodes in ” , which is a contraction with the
7
S
S
observation that it belongs to hbœ .
A note on domino treewidth
143
As for hbœ , each case can appear only once, hbœ is adjacent to at most two nodes in ” .
7
To show that aŸ” a –f—Z˜ja•” a6MnJ , build a tree ¡ in the following way: ” is the set of nodes in ¡ . If
7
7
S
yv@” has an ancestor that also belongs to ” , then take an edge from y to the closest ancestor that also
7
7
belongs to ” . One can observe that ¡ is indeed a tree. Every node ym@A” M#” must have at least two
7
7
S
children. So ¡ is a tree with at most aŸ” a leaves, and without nodes with one child. A well know fact
S
about trees tells us that ¡ has at most —HaŸ” a>MOJ nodes, hence a•” a–O—™˜:a•” a>MOJ . ¢
S
7
S
Lemma 2.2 Let (£[]`_IaTcd@oejg6*-h¤'¥( ej*<kU.G. be a tree decomposition of &¥'¦( )+*-,". . Let ˆ
a ˆ¨a'nš . Let ©EªOš .
~§)
,
1. There exists a set of «š¬( ©™DfJL.z­ nodes ” †
~ e , such that each connected component of &` )M
S
q
contains
at
most
vertices
from
.
`
]
4
_
‘
©
ˆ
 r6®¯
2. There exists a set of — «š¬(©EDJ^.\­°MfJ nodes ”
~¤e , such that each connected component ±
7
of &` )†M q  r6®² ] _ ‘ contains at most © vertices from ˆ , and for each connected component of
q
&` )fM
 r6®² ] _ ‘ there are nodes c S *-c 7 @³” 7 , such that every vertex ? that is adjacent to a vertex
] _´¯
] _² .
in ± belongs to ±
›
›
Proof: 1. First, observe that for any ”’~†e , &` )M q  r6® ]`_‘ consists of a number of connected
components, such that for any connected component ± of &` )xM q  r6® ]`_‘ , we have a subtree ”:µ of the
forest hZ eUMO”¶‘ with ±ƒ~ q  r6®· ]  , i.e., removing ” from e splits h in a number of disjoint trees, and
each connected component has its vertices in the bags of the nodes in only one of these subtrees.
Choose an arbitrary root š°@{e , and view h as a rooted tree. We will process h in a bottom-up order: a
node is processed after all its children are processed. While processing vertices, we maintain a set ” ~Ae ,
S
which is initially empty, and a set ˆxœ~Oˆ , for which initially ˆxœ¶'nˆ . The idea is that nodes are added
to ” until finally the requested set is found, and that ˆxœ gives those vertices in ˆ that still can belong to
S
a connected component with too many vertices in ˆ in it.
For a node cY@me , let )j_l' q  rsz¸ ]`_ , with e_ the set containing c and all its descendants in h .
While processing a node c , compute ¹H_'†)H_ | ˆxœ . If a ¹w_<a ºƒ© , put c in ” , and set ˆxœY'¨ˆ»MA¹H_ .
S
Otherwise, nothing is done when processing node c .
We now claim that the set ” which is obtained after processing root node š fulfils the requirements
S
of the lemma. Consider a connected component ± of &` )¼M q  r6®^¯ ]`_‘ . Let c£µ be the highest node
q
in h whose bag contains a vertex in ± . Clearly, c£µ¾@
½ ” , as ±¿~§)¨M
 r6®¯ ]`_ . Hence, when c\µ
S
was processed, a ¹H_·ÀaÀ–Á© . Now we note that ± | ˆ
~¨¹H_· : suppose ?O@’± | ˆ . By choice of c£µ ,
or in ” , and hence
?f@“)j_· . If ?f@ˆÂM’ˆxœ , then ? belongs to a bag that is below a node in ”
S
S
either ? belongs to q  r6® ¯ ]`_ or is separated from ± by q  r6® ¯ ]`_ . This contradicts that ?@K± , hence
± | ˆ~n± | ˆxœÃ~A¹ _ · , and we have a ± | ˆ¨aT–Ba ¹ _ ·ÀaR–K© .
To each node cY@” , we can associate the ©lDÄJ or more vertices that are removed from ˆxœ when c was
S
added to ” . As each vertex in ˆ is associated with at most one cY@” , we have a•” aR–†«š¬( ©DnJ^.z­ .
S
S
S
2. First, obtain a set ” as above. As in Lemma 2.1 on the preceding page, create set ” , such that
7
S
p
p
a•”
p
”
S
7
a–#— «š¬(©‚DnJ^.z­MOJ
~’”
S
/
.
Every subtree of hZ eQM” ‘ is adjacent to at most two nodes in ” .
7
7
144
Hans L. Bodlaender
Now, let ± be a connected component of &` )BM q  r6®² ]  ‘ . By the properties of tree decompositions,
it follows that there is a subtree hõ of hZ eEMA” ‘ , such that all vertices of ± only belong to bags of nodes
7
in hõ . Thus, the vertices that neighbour a vertex in ± but do not belong to ± must belong to a bag ]1_ ,
with c‚@‡” one of the at most two nodes in ” that are adjacent to hõ . ¢
7
7
Corollary 2.3 Let &Å'Å()Y*<,Z. have treewidth at most $ and maximum degree at most % . Let ˆÆ~ƒ) ,
a ˆ¨aR–Oš . Let ©Uªš . There exists a set ÇA~#) of at most ( $bDAJL.˜>( — «š¬(©NDAJL.z­€MÄJ^. vertices, such every
connected component ± in &` )xM³Çl‘ contains at most © vertices from ˆ and at most (—$EDO—.G% vertices
that are adjacent to a vertex in Ç . If $ is a constant, such a set Ç can be found in linear time.
Proof: The non-algorithmic result follows directly from the previous lemma. (Note that for such a
component ˆ , there are at most —$°Dn— vertices in Ç adjacent to vertices in ˆ (namely the vertices in
at most two bags of the tree decomposition), hence at most (—$ZD#—6.£% vertices in ˆ that are adjacent to
a vertex in Ç .) To effectively obtain the set Ç , first apply the algorithm in [Bod96] to obtain an arbitrary
tree decomposition of width at most $ . It is not hard to see that the proofs given above then can be carried
out in linear time. ¢
3 The domino treewidth theorem
In this section, we prove the main result of this section. The technique is inspired by a technique from
[BGHK95], which was again inspired by a technique from [RS95].
Theorem 3.1 Let &o'ƒ( )+*-,Z. be a graph with treewidth at most $ and maximum degree at most % . Then
the domino treewidth of & is at most ( C6$EDAF.£%w(4%QDKJL.lMAJ .
Proof:
We first give a recursive procedure, called MAKEDEC, called with two arguments: a graph È
'
( )HÉ"*<,ÉQ. (which is always an induced subgraph of & , and is assumed to have treewidth at most $ ,
and maximum degree at most % ), and a set of vertices ˆ‰~t) É . The procedure outputs a rooted domino
tree decomposition of È , (\[]_ œ ac+@ve6œ4g6*-hbœj'o( e6œ*<kEœÊ.-. of width at most (4C:$IDÄF.£%w(4%bDAJ^.ËM³J , such that
the vertices in ˆ only belong to the bag of the root node of the domino tree decomposition.
Procedure MAKEDEC (graph Ⱦ'ƒ( ) É *-, É . , vertex set ˆ ) has the following steps:
1. Obtain a set ÇA~K)jÉ , such that every connected component of ȳ )jÉOMÇl‘ contains at most =:$™DŒ—
vertices from ˆ and at most ( —6$`Dt—6.£% vertices that are adjacent to a vertex in Ç , (as in Corollary 2.3.)
2. Set Ìx'
Š
( Ç
›
ˆf.
.
3. Compute the connected components È
S
'o( )
S
*<,
S
.
, . . . , È1Í'ƒ( )wÍL*-,bÍ. of ȳ )HÉKMŒÇMĈ’‘ .
4. For each c , JE–AcY–K© , call MAKEDEC( È1_ , )j_ | Ì ).
5. Combine the tree decompositions obtained in the previous step in the following way: Take a new
Ç
ˆ . This is the root of the new tree decomposition. Make š adjacent to
node š with ]1Îb'tÌ
›
›
the roots of each of the tree decompositions, obtained in the previous step. The result is the output
of the procedure.
A note on domino treewidth
145
Assume that the set Ç found in Step 1 is at most of the size, guaranteed to exist by Corollary 2.3 on the
preceding page, i.e., we have:
a ÇaR–x( $EDnJ^.N˜(—Ë«
š
©‚DnJ
­MOJ^.
Claim 3.1.1 Let Èi'“( )HÉZ*<,ə. be a connected graph, and ˆ‰~#)HÉ , ˆÏ't
½ Ð . When MAKEDEC( Èm*ˆ )
is called, the procedure outputs a rooted domino tree decomposition of È , such that vertices in ˆ only
belong to the root bag of the domino tree decomposition.
Proof: First, observe that the first parameter of a recursive call to MAKEDEC always is a connected
graph, and the second parameter of every recursive call to MAKEDEC is always a non-empty set: every
ˆ . Thus, the recursive
connected component of ȳ ) É M³ÇMĈ’‘ must contain vertices adjacent to Ç
›
calls done to MAKEDEC involve graphs with fewer vertices, hence the procedure terminates.
ˆf.E'x
½ Ð , then Ñ and ? belong both to the root bag ] Î . Otherwise,
Let [Ñl*G?Hg"@‡,bÉ . If [Ñl*G?Hg | ( Ç
›
Ò
and Ó belong to the same connected component È1_ of ȳ )HÉOM‡ÇvM‡ˆ’‘ , and by induction, there will be
a bag containing both Ò and Ó . In both cases, there is a bag in the resulting decomposition that contains
both Ò and Ó .
Let ?d@v)HÉ . There are three cases.
ˆ , then ? does not belong to any connected component of ȳ ) É M³ÇMĈ’‘ , hence ? only
If ?d@Ç
›
belongs to bag ]1Î , and no other bag of the decomposition.
If ?K@’Ì , then ? belongs to ] Î . In addition, ? belongs to exactly one connected component È1_ of
ȳ )HÉBM³ÇŒMˆ’‘ . By induction, ? belongs to the root bag of the domino tree decomposition yielded by
the call of MAKEDEC( È1_ , )H_ | Ì ) and no other bag. Thus, ? belongs to exactly two bags that are adjacent.
½ Ì
Ç
ˆ , then ? belongs to exactly one connected component È _ of ȳ ) É MtÇOMtˆ’‘ ,
If ?“@ƒ
›
›
and by induction to one or two adjacent bags in the decomposition made by the recursive call to
MAKEDEC( È _ , ) _H| Ì ). ? does not belong to any other bag.
Hence, the claim follows. ¢
Claim 3.1.2 If MAKEDEC( Èm*<ˆ ) is called with ÈX'Ô( ) É *<, É . a connected graph of maximum degree
% and treewidth at most $ , and ˆ
~†)HÉ a set of vertices of size at most (436$1DO=R.£% , then the resulting
domino tree decomposition has width at most (4C:$QDOF!.G%H( %™DnJ^.lMOJ .
Proof: First, we estimate the size of the root bag of the resulting domino tree decomposition. We
have a ˆ¨aR–B(43:$QD³=:.G% . By Corollary 2.3 on the page before, we can take:
a ÇaR–x( $UDnJ^.(—Ë«G(436$QD³=:.£%¬(4=:$EDÕ6.z­MOJ^.€ªOÕj($UDKJL.£%
Now
a Ì`a¿–
%U˜6a Ç
ˆ¨a
›
%U˜(-(43:$UDÄ=R.£%IDÕj($QDnJ^.G%:.
–
'
(4C:$QDOF!.G%H( %™DnJ^.
So,
aÌ
›
Ç
›
ˆ¨aR–’( C6$EDOF!.£%w(4%QDKJL.
146
Hans L. Bodlaender
Secondly, we estimate the size of a set )j_ | Ì in a recursive call MAKEDEC( È1_ , )j_ | Ì ). Write
) _j|
Ìx'o() _j|
Š
(ÇY.G.
›
Š
() _w|
( ˆf.G.
È _ œ of ȳ ) É M
Each connected component È _ of ȳ ) É MdÇ1Mdˆ’‘ is contained in a connected component
Š
Çl‘ . È_ œ contains at most =R$ZD#— vertices from ˆ , hence at most (=:$"DK—.G% vertices of
( ˆf. . Also, by
Š
construction of Ç , È_ œ contains at most (—$EDA—.£% vertices in ( ÇY. . As a consequence,
a ) _j|
Ì`aR–x(=:$EDA—.G%IDn(—$EDA—.£%°'o( 36$EDÄ=R.£%
Now, we can use induction: each recursive call of MAKEDEC is called with as second parameter a set of
size at most ( 36$…D‡=:.G% , hence the recursive calls give tree decompositions of width at most (4C6$…DŒF!.G%H( %€D
J^.lMOJ , which proves the claim. ¢
So, from these two claims it follows, that when we call MAKEDEC( & , ˆ ) for a connected graph & of
treewidth at most $ , and maximum degree at most % , and any non-empty vertex subset ˆ which has size
at most ( 36$EDÄ=R.£% , we obtain a domino tree decomposition of & of width at most ( C6$EDOF!.£%w(4%QDKJL.lMAJ .
If & is not connected, then make separate domino tree decompositions for each connected component,
and connect these to a tree in an arbitrary way. ¢
The new idea in the proof can be found in step 2 of the procedure MAKEDEC: by adding the neighbours
of the vertices in set Ç
to the root bag of the tree decomposition to make, we do not have to use
ˆ
›
these vertices at lower levels of the tree decomposition anymore. Apart from this idea, the structure of the
algorithm is similar to algorithms found in [RS95, BGHK95].
Corollary 3.2 Let $ be a constant. Given a graph with treewidth at most $ and maximum degree at most
% , a domino tree decomposition of & of width at most ( C6$EDOF!.£%w(4%QDKJL.lMAJ can be built in Ö1(4×Ã7^. time.
Proof:
Use the procedure, given in the proof above. Excluding the time spent in recursive calls of
one call of MAKEDEC uses Ö1(4×Ë. time. There are Ö1(4×Ë. such calls (e.g., every vertex belongs
to at most two bags, hence a tree decomposition with Ö1(4×Ë. nodes is obtained, and the number of recursive
calls of MAKEDEC equals the number of nodes of the resulting tree decomposition), so the total time is
bounded by Ö1(×Ã7L. . ¢
MAKEDEC,
4 A lower bound
In this section, we show that a general bound like obtained in the previous section must always be of order
PQ( $%R. .
We first start with the following lemma, which is also interesting on its own. For a graph &“'( )+*-,Z. ,
let
Ò
& 7 'o( )+*[[^?H*-uEgUa>[^?H*-uEgU@m,#ØvŽ
@v)[^?H*
Ò
gU@v,nه[
Ò
*-uEgU@v,1gL.
Lemma 4.1 Let &f'ƒ( )+*-,". be a graph with domino treewidth at most $ . The treewidth of &E7 is at most
—$ .
Proof: W.l.o.g., suppose & is connected. Let (\[^] _ a8c‚@{ejg6*-hK'o(4eH*-kU.G. be a domino tree decomposition of & of width at most $ . Note that (by the properties of tree decompositions and the assumption of
connectedness of & ) each two adjacent bags intersect. Choose an arbitrary root š . If we add to each bag
A note on domino treewidth
147
Fig. 1: Grid with added vertices ÚÛ
]
the bag of the parent of c (unless c'ƒš ), then we obtain a tree decomposition of & of width at most
. (The union of two bags with a non-empty intersection and with size at most $QDnJ each is taken.)
For every edge [?w*GuUg in &E7 , we have a bag containing both ? and u : this is trivially true if [^?H*-uEgU@v, .
If ? and u have a common neighbour Ò in & , then either there is a bag ] _ containing both ? , u , Ò , or
there are two adjacent bags, one containing ? and Ò , and one containing u and Ò . One must be a child in
Ò
h (with root š ) of the other. Thus, ? , u , and all three belong to a common bag in the constructed tree
decomposition. ¢
_
—$
A J>¬>ÕUM#—6¬!Õ -separator of a set W in a graph &Å'¾()Y*<,Z. , is a set of vertices Ç , such that ˆ can be
partitioned into sets ˆ , ˆ , and ˆ , with ˆ
'fÇ | ˆ , a ˆ
aw–B—:¬>Õwa ˆ¨a , a ˆ
aw–’—:¬>ÕHa ˆ¨a , and every
7
9
9
7
S
S
path from a vertex in ˆ
to a vertex in ˆ
uses a vertex in Ç .
7
S
The following lemma is well known. See e.g. [BGHK95, GRE84, Liu90, RS86].
Lemma 4.2 Let &Ü'Ý()Y*<,Z. be a graph of treewidth at most $ . Let ˆ
of size at most $QDnJ .
JL¬>ՙMė6¬!Õ -separator of ˆ
Lemma 4.3 For all %³Þ¼; , $Oކ— , $ even, there exists a graph &
degree at most % , and domino treewidth at least S $%ZMŒ— .
~X)
. Then &
contains a
with treewidth at most $ , maximum
S 7
Proof: Consider the following graph.
First, we take a grid of size $j¬— by %67^$ . I.e., we have vertices of the form ?!_ß  , J’–àcŒ–¥$j¬— ,
JZ–³yd–n%67L$ , and ?!_ß  is adjacent to ?!_Êá4ß  á , iff a c MŒc\œzaDBa yZMy:œa'ƒJ . To this grid, we add $H¬!— additional
vertices â *ããã*-â }ä , with, for each c , Jd–fc™–ƒ$H¬!— , â _ adjacent to each vertex ? _4ß å æ8} , Jd–ty–o% . Let
S
7
&x'ƒ( )+*-,Z. be the resulting graph.
See Figure 1 for an illustration of the construction. (In order to make the figure not too large, the
distance between successive neighbours of the vertices âL_ is 4 in the figure, instead of %R$ .)
The maximum degree of & is /`0>2 ( ;T*-%R. : vertices of the form ?!_ß  have degree at most five, while
vertices of the form âL_ have degree % . It is also not hard to see that the treewidth of & is at most $ . The
$j¬!— by %67^$ grid graph has treewidth exactly $j¬— (see e.g. [Bod98].) As & contains $j¬!— vertices such that
when these are deleted from & , & becomes a graph of treewidth $j¬!— , the treewidth of & is at most $ . (See
e.g. [Bod98], Lemma 72.)
Call the c th row the set of all vertices of the form ?!_4ß  , J°–Oyv–B% 7 $ . Similar, the set of all vertices of
the form ?!_4ß  , JE–OcY–K$j¬!— is called the y th column.
Now, we claim that &E7 has treewidth at least çS %R$`MtJ . Note that all vertices in the c th row that were
148
Hans L. Bodlaender
adjacent to âL_ form a clique in &E7 . Call the set of these vertices the c th row-clique. Let
ˆ»'x[^? _ß ™a6JE–c‚–#$H¬!—*+JE–y`–K$% 7 g
I.e., ˆ is the set of the grid vertices in & .
Suppose Ç is a J>¬>ՋMx—6¬!Õ -separator of ˆ of minimum size in &E7 , partitioning ˆ into ˆ , ˆ ,
7
S
ˆ
'tÇ | ˆ .
9
We will now show that a ÇaRÞ çS %R$ . Assume a Ça:ª çS %R$ .
aRÞ¨çS %:7$7 , and likewise a ˆ
aކçS %67L$7 .
Note that a ˆ a– 7 ˜^%67^$7^¬— , hence a ˆ
| ˆ
| ˆ
7
9
9
S
S
9
Every column that contains both a vertex in ˆ
and a vertex in ˆ
must also contain a vertex in Ç .
7
S
Thus, we may assume there are fewer than çS %R$ such columns. So, fewer than S %R$7 vertices in ˆ can
belong to such a column. It follows that there are at least ( çS %67L$7wM S %R$7^.-¬($j¬!—6.ÀS 7 ' S %67^$YM çS %R$ columns
that only contain vertices in ˆ , and thus, every row contains S %67^$ÀS M 7 çS %:$ vertices in 9 ˆ . Likewise, every
S
S
9
row contains S %:7^$"MoçS %R$ in ˆ .
7
9
We now will show that every row contains at least S %R$ vertices in Ç .
9
Consider the c th row. Note that either all vertices in the c th row-clique belong to ˆ
Ç or all vertices
SN›
in the c th row-clique belong to ˆ
Ç . Without loss of generality, we suppose the former; the other case
7 ›
is identical.
We partition the vertices in the c th row in % intervals, where the è th interval contains vertices
? _ߟéëêIì
S % of these intervals must
æ8}î *-? _ߟéÊêbì
æ8}8î *ããã*G?!_4ß ê æ8} . At least ï£( S %:7$vM çS %:$j.-¬( %:$j.zð³Þ
7
S£í
S
S£í
9
contain vertices in ˆ . However, each interval also contains
a vertex in the c th9 row-clique, hence it con7
tains a vertex in Ç
ˆ
. So, each interval that contains a vertex in ˆ
must contain a vertex in Ç , hence
7
›
S
the c th row contains at least S % vertices in Ç .
9
As we have $j¬!— rows, it follows that a ÇaHÞ¾çS %:$ . By Lemma 4.2 on the page before, we have that the
treewidth of &E7 is at least çS %:$‹MnJ , hence by Lemma 4.1 on page 146, the domino treewidth of & is at
least S %R$°Mė . ¢
S 7
5 Final remarks
It is possible to give a modified version of the procedure of Corollary 3.2 on page 146, that yields domino
tree decompositions of somewhat larger width (but still of Ö1( $%:7>. , but that uses Ö1(×Qñëò6ó+×Ë. time instead
of Ö1(×Ã7L. time. However, the proof in [BE97] can be turned into an algorithm that uses linear time. It is
not known how much time a procedure based upon the proof by Ding and Oporowski [DO95] would take.
The proof given in this paper seems unable to yield linear time algorithms - the approach typically leads
to algorithmic results of PQ(×Qñëò6óY×Ë. time. It is open whether domino tree decompositions of Ö1($%67>. width
can be obtained with a linear time algorithm.
Another interesting open problem is whether a bound of Ö1($ 7 %:. can essentially be improved. It would
be interesting to see if better bounds, e.g., a bound of Ö1( $%R. can be proved, and whether better lower
bounds are possible.
In some special cases, better bounds can be obtained. For instance, for trees we have the following easy
result.
Theorem 5.1 The domino treewidth of a tree is at most its maximum degree.
A note on domino treewidth
149
Proof: Let h be a tree with maximum degree % . Choose an arbitrary root š , and view h as a rooted
tree. Let hbœ¶'( )Uœ *-,Zœ´. be the tree, obtained by removing all leaves from h . Consider the following tree
decomposition of h : (£[]`ôQa?õ@m)Uœ4g6*-hbœë. , where each set ]`ô consists of ? and all children of ? in h . One
easily verifies that this is a domino tree decomposition of h with width at most % . ¢
So for trees (and similarly for forests), the domino treewidth is linear in its degree. (Note also that the
domino treewidth of a graph with maximum degree %vޓJ is at least ï£( %QD’JL.-¬!—>ðbM#J : at most two bags
can contain a vertex of degree % and all its neighbours.) It seems interesting to see if it is also possible
to obtain similar bounds for other restricted classes of graphs of bounded treewidth, e.g., series parallel
graphs, Halin graphs, or arbitrary graphs of treewidth two.
References
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Hans L. Bodlaender and Joost Engelfriet. Domino treewidth. J. Algorithms, 24:94–123,
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Guoli Ding and Bogdan Oporowski. Some results on tree decomposition of graphs. J. Graph
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150
Hans L. Bodlaender