International audienceTree-decompositions are the cornerstone of many dynamic programming algorithms for solving graph problems. Since the complexity of such algorithms generally depends exponentially on the width (size of the bags) of the decomposition, much work has been devoted to compute tree-decompositions with small width. However, practical algorithms computing tree-decompositions only exist for graphs with treewidth less than 4. In such graphs, the time-complexity of dynamic programming algorithms is dominated by the size (number of bags) of the tree-decompositions. It is then interesting to minimize the size of the tree-decompositions. In this extended abstract, we consider the problem of computing a tree-decomposition of a graph with width at most k and minimum size. We prove that the problem is NP-complete for any fixed k ≥ 4 and polynomial for k ≤ 2; for k = 3, we show that it is polynomial in the class of trees and 2-connected outerplanar graphs

Li, Bi

Nisse, Nicolas

Suchan, Karol

Zahra Moataz, Fatima

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Minimum Size Tree-decompositions
Bi Li, Fatima Zahra Moataz, Nicolas Nisse, Karol Suchan
To cite this version:
Bi Li, Fatima Zahra Moataz, Nicolas Nisse, Karol Suchan. Minimum Size Tree-decompositions.
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2015, Beberibe, Ceara´, Brazil. <hal-01162695>
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Minimum Size Tree-decompositions 1
Bi Li a,b,c,2, Fatima Zahra Moataz b,a,3, Nicolas Nisse a,b,4 and
Karol Suchan d,e,5
a Inria, France
b Univ. Nice Sophia Antipolis, CNRS, I3S, UMR 7271, Sophia Antipolis, France
c Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing, China
d FIC, Universidad Adolfo Iba´n˜ez, Santiago, Chile
e WMS, AGH, University of Science and Technology, Krakow, Poland
Abstract
Tree-decompositions are the corner-stone of many dynamic programming algorithms
for solving graph problems. Since the complexity of such algorithms generally de-
pends exponentially on the width (size of the bags) of the decomposition, much work
has been devoted to compute tree-decompositions with small width. However, prac-
tical algorithms computing tree-decompositions only exist for graphs with treewidth
less than 4. In such graphs, the time-complexity of dynamic programming algo-
rithms is dominated by the size (number of bags) of the tree-decompositions. It is
then interesting to minimize the size of the tree-decompositions. In this extended
abstract, we consider the problem of computing a tree-decomposition of a graph with
width at most k and minimum size. We prove that the problem is NP-complete for
any fixed k ≥ 4 and polynomial for k ≤ 2; for k = 3, we show that it is polynomial
in the class of trees and 2-connected outerplanar graphs.
Keywords: Minimum size tree-decomposition, treewidth, NP-hard.
1 Introduction
A tree-decomposition of a graph [9] is a way to represent it by a family of
subsets of its vertex-set organized in a tree-like manner and satisfying some
connectivity property. More formally, a tree decomposition of a graph G =
(V,E) is a pair (T,X ) where X = {Xt|t ∈ V (T )} is a family of subsets of V ,
called bags, and T is a tree, such that: (1)
⋃
t∈V (T )Xt = V ; (2) for any edge
uv ∈ E, there is t ∈ V (T ) such that Xt contains both u and v; and (3) for
any vertex v ∈ V , the set {t ∈ V (T )|v ∈ Xt} induces a subtree of T .
The width of (T,X ) is maxt∈V (T )|Xt|− 1 and its size is the order |V (T )| of
T . If T is a path, (T,X ) is called a path-decomposition of G. The treewidth
(resp., pathwidth) of G, denoted by tw(G) (resp., pw(G)), is the minimum
width over all tree-decompositions (resp., path-decompositions) of G.
Tree-decompositions are the corner-stone of many dynamic programming
algorithms for solving graph problems. For example, the famous Courcelle’s
Theorem states that any problem expressible in MSOL can be solved in linear-
time in the class of bounded treewidth graphs [5]. Another framework based on
graph decompositions is the bi-dimensionality theory that allowed the design
of sub-exponential-time algorithms for many problems in the class of graphs
excluding some fixed graph as a minor (e.g., [6]). Given a tree-decomposition
with width w and size n, the time-complexity of most of such dynamic pro-
gramming algorithms can often be expressed as O(2wn) or O(2w logwn). There-
fore, the problem of computing tree-decompositions with small width has
drawn much attention in the last decades.
The above-mentioned algorithms have mainly theoretical interest because
their time-complexity depends exponentially on the treewidth and, on the
other hand, no practical algorithms are known to compute a good tree-de-
composition for graphs with treewidth at least 5. However, for small (≤ 4)
treewidth graphs, practical algorithms exist to compute tree-decompositions
with minimum width [11,2,10]. Since the computation of tree-decompositions
is a challenging problem, we propose in this paper to study it from a new
point of view. Namely, we aim at minimizing the number of bags of the tree-
1 This work has been partially supported by ANR project Stint (ANR-13-BS02-0007), the
associated Inria team AlDyNet, the project ECOS-Sud Chile and a grant from the ”Conseil
re´gional Provence Alpes-Coˆte d’Azur”.
2 Email: bi.li@inria.fr
3 Email: fatima-zahra.moataz@inria.fr
4 Email: nicolas.nisse@inria.fr
5 Email: karol.suchan@uai.cl
decomposition when the width is bounded. This new perspective is interesting
on its own and we hope it will allow to gain more insight into the difficulty of
designing practical algorithms for computing tree-decompositions.
We consider the problem of computing tree-decompositions with mini-
mum size. If the width is not constrained, then a trivial solution is a tree-
decomposition of the graph with one bag (the full vertex-set). Hence, given
a graph G and an integer k ≥ tw(G), we consider the problem of minimizing
the size of a tree-decomposition of G with width at most k.
Related Work. The problem of computing “good” tree-decompositions has
been extensively studied. It is NP-hard to compute tree-decompositions with
minimum width [1]. For any fixed k ≥ 1, Bodlaender designed an algorithm
that computes, in time O(kk
3
n), a tree-decomposition of width k of any n-
node graph with treewidth at most k [3]. Recently, a single-exponential (in k)
algorithm has been proposed that computes a tree-decomposition with width
≤ 5k in the class of graphs of treewidth ≤ k [4].
We are not aware of any work dealing with the computation of tree-
decompositions with minimum size. In [7], Dereniowski et al. study the
problem of size-constrained path-decompositions. Given any k > 0 and any
graph G with pathwidth at most k, let lk(G) denote the smallest size (length)
of a path-decomposition of G with width at most k. For any fixed k ≥ 4 (resp.
k ≥ 5), computing lk is NP-complete in general graphs (resp., in the class of
connected graphs) [7]. Moreover, computing lk can be solved in polynomial-
time in the class of graphs with pathwidth at most k for any k ≤ 3. The
dual problem is also hard: for any fixed l ≥ 2, it is NP-complete in general
graphs to compute the minimum width of a path-decomposition with length l
[7]. Moreover, by the same techniques, the original proof can be extended to
an analogous result on treewidth.
Our results. For any tw > 0 and any graph G of treewidth at most tw, let
sk(G) denote the minimum size of a tree-decomposition of G of width at most
k, k ≥ tw. We consider the complexity of computing sk for k ≥ 1. Table 1
summarizes our results as well as the remaining open questions.
s1 s2 s3 s4 sk, k = max{tw + 1, 5}
Graphs of treewidth at most tw = 1 P (trivial) P P ? ?
Graphs of treewidth at most tw = 2 - P ? ? ?
Graphs of treewidth at most tw = 3 - - ? NP-hard ?
Graphs of treewidth at most tw ≥ 4 - - - NP-hard NP-hard
Table 1
2 Complexity results on the computation of sk
In this section, we present some NP-hardness results and some polynomial
algorithms for the computation of sk for k ≥ 1. Due to lack of space, all
proofs have been omitted and can be found in [8]. First we prove that:
Theorem 2.1 For any fixed integer k ≥ 5, the problem of computing sk is
NP-complete in the class of connected graphs with treewidth at most k − 1;
furthermore, the problem of computing s4 is NP-complete in the class of planar
graphs with treewidth at most 3.
Our proof extends the one of [7] for size-constrained path-decompositions.
In the rest of this extended abstract, we consider the computation of sk
for k ≤ 3. We give polynomial-time algorithms in several graph classes. Let
us first present the general approach that is used in what follows.
General Approach. Let k ≥ 1 and G be a graph with tw(G) ≤ k. A subset
B ⊆ V (G) is a k-potential-leaf if there is a tree-decomposition (T,X ) with
width at most k and size sk(G) such that B is a leaf bag of (T,X ) (i.e., the
node corresponding to bag B in T is a leaf). Abusing the notations, we will
identify the subset B with the subgraph it induces in G. Given a class of
graphs C and a positive integer k, a set of graphs P is called a complete set
of k-potential-leaves of C, if for any graph G ∈ C, there exists a graph H ∈ P
such that G contains a k-potential leaf isomorphic to H.
The key idea of our algorithms is to identify a complete set of potential-
leaves. Then we proceed recursively: we find in G a k-potential-leaf H from
the complete set, put it in a bag and combine in with the minimum size
tree-decomposition of G \H. The next lemmas formalize this idea.
Given a graph G = (V,E) and a subset S ⊆ V , we denote by GS the graph
obtained from G by adding the minimum number of edges to G such that S
induces a clique in GS.
Lemma 2.2 Let k ≥ 1 and G = (V,E) be a graph with tw(G) ≤ k. Let
B ⊆ V be a k-potential-leaf of G. Let S ⊂ B be the set of vertices of B that
have a neighbor in V \B. Then sk(G) = sk(GS \ (B \ S)) + 1.
This lemma implies the following corollary:
Corollary 2.3 Let k ∈ N∗ and C be the class of graphs with treewidth at most
k. If there is a polynomial-time algorithm that, for any graph G ∈ C, computes
a k-potential-leaf of G, then sk and a minimum size tree decomposition of width
at most k can be computed in polynomial-time in C.
From Corollary 2.3, solving the Minimum Size Tree-Decomposition prob-
lem in a class of graphs C reduces to the problem of finding a k-potential-leaf
for any graph G ∈ C. This can be done in polynomial time if we characterize
a finite complete set of k-potential-leaves of C. In the rest of this section, we
present our results for k ∈ {2, 3}.
Minimum Size Tree-Decompositions for k ∈ {2, 3}. We first charac-
terize a complete set of 2-potential-leaves in the class of non-empty graphs
(containing at least one edge) of treewidth at most 2 (see Fig. 1). Then, we
identify a complete set of 3-potential-leaves in the class of trees with at least
four vertices and in the class of 2-connected outerplanar graphs (see Fig. 2).
Note that the complete sets of potential-leaves in the figures are finite and
each potential-leaf in them can be found in polynomial time. This together
with Corollary 2.3 imply the following result:
q
p
q
bf
c
a
c a
ffp bα
Fig. 1. Complete set of 2-potential-leaves of graphs of treewidth at most 2
g
g
h
p
p
f
g
f
p
'f
g
f
p
'
f
p
f f '
f ''f
(a) Trees
a
e
dcb
a dcb
x ya
dcb
(b) 2-connected outerplanar graphs
Fig. 2. Complete sets of 3-potential-leaves of trees and 2-connected outerplanars
Theorem 2.4 s2 and a minimum size tree decomposition can be computed in
polynomial-time in general graphs; s3 and a minimum size tree decomposition
of width at most 3 can be computed in polynomial-time in the class of trees
and 2-connected outerplanar graphs.
3 Conclusion
In this extended abstract, we give preliminary results on the complexity of
minimizing the size of tree-decompositions with given width. Table 1 sum-
marizes our results as well as the remaining open questions. We currently
investigate the computation of s3 in the class of graphs with treewidth 2 and
sk for k ≥ 3 in the class of trees. These cases are more intricate than the
polynomial cases we have considered. It seems that a global view of the graph
needs to be considered to decide whether a subgraph is a 3-potential leaf of the
graph; in addition, a k-potential leaf of a tree for k ≥ 5 can be disconnected
(illustrating examples can be found in [8]).
References
[1] Arnborg, S., D. G. Corneil and A. Proskurowski, Complexity of finding
embeddings in a k-tree, SIAM J. Algebraic Discrete Methods 8 (1987), pp. 277–
284. URL http://dx.doi.org/10.1137/0608024
[2] Arnborg, S. and A. Proskurowski, Characterization and recognition of partial
3-trees, SIAM J. Algebraic Discrete Methods 7 (1986), pp. 305–314. URL http:
//dx.doi.org/10.1137/0607033
[3] Bodlaender, H. L., A linear-time algorithm for finding tree-decompositions of
small treewidth, SIAM J. Comput. 25 (1996), pp. 1305–1317.
[4] Bodlaender, H. L., P. G. Drange, M. S. Dregi, F. V. Fomin, D. Lokshtanov
and M. Pilipczuk, A o(ckn) 5-approximation algorithm for treewidth, CoRR
abs/1304.6321 (2013).
[5] Courcelle, B., The monadic second-order logic of graphs. i. recognizable sets
of finite graphs, Information and Computation 85 (1990), pp. 12 – 75. URL
http://www.sciencedirect.com/science/article/pii/089054019090043H
[6] Demaine, E. D. and M. Hajiaghayi, The bidimensionality theory and its
algorithmic applications, Comput. J. 51 (2008), pp. 292–302.
[7] Dereniowski, D., W. Kubiak and Y. Zwols, Minimum length path
decompositions, CoRR abs/1302.2788 (2013).
[8] Li, B., F. Z. Moataz, N. Nisse and K. Suchan, Size-Constrained Tree
Decompositions, Research report, INRIA Sophia-Antipolis (2014). URL https:
//hal.inria.fr/hal-01074177
[9] Robertson, N. and P. D. Seymour, Graph minors. ii. algorithmic aspects of
tree-width, J. Algorithms 7 (1986), pp. 309–322.
[10] Sanders, D. P., On linear recognition of tree-width at most four, SIAM J. Discret.
Math. 9 (1996), pp. 101–117.
[11] Wald, J. A. and C. J. Colbourn, Steiner trees, partial 2-trees, and minimum ifi
networks, Networks 13 (1983), pp. 159–167.


Minimum Size Tree-decompositions

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HAL Id: hal-01162695https://hal.inria.fr/hal-01162695Submitted on 11 Jun 2015HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.Minimum Size Tree-decompositionsBi Li, Fatima Zahra Moataz, Nicolas Nisse, Karol SuchanTo cite this version:Bi Li, Fatima Zahra Moataz, Nicolas Nisse, Karol Suchan. Minimum Size Tree-decompositions. LA-GOS 2015 – VIII Latin-American Algorithms, Graphs and Optimization Symposium, May 2015,Beberibe, Ceará, Brazil. ￿hal-01162695￿Minimum Size Tree-decompositions 1Bi Li a,b,c,2, Fatima Zahra Moataz b,a,3, Nicolas Nisse a,b,4 andKarol Suchan d,e,5a Inria, Franceb Univ. Nice Sophia Antipolis, CNRS, I3S, UMR 7271, Sophia Antipolis, Francec Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing, Chinad FIC, Universidad Adolfo Ibáñez, Santiago, Chilee WMS, AGH, University of Science and Technology, Krakow, PolandAbstractTree-decompositions are the corner-stone of many dynamic programming algorithmsfor solving graph problems. Since the complexity of such algorithms generally de-pends exponentially on the width (size of the bags) of the decomposition, much workhas been devoted to compute tree-decompositions with small width. However, prac-tical algorithms computing tree-decompositions only exist for graphs with treewidthless than 4. In such graphs, the time-complexity of dynamic programming algo-rithms is dominated by the size (number of bags) of the tree-decompositions. It isthen interesting to minimize the size of the tree-decompositions. In this extendedabstract, we consider the problem of computing a tree-decomposition of a graph withwidth at most k and minimum size. We prove that the problem is NP-complete forany fixed k ≥ 4 and polynomial for k ≤ 2; for k = 3, we show that it is polynomialin the class of trees and 2-connected outerplanar graphs.Keywords: Minimum size tree-decomposition, treewidth, NP-hard.1 IntroductionA tree-decomposition of a graph [9] is a way to represent it by a family ofsubsets of its vertex-set organized in a tree-like manner and satisfying someconnectivity property. More formally, a tree decomposition of a graph G =(V,E) is a pair (T,X ) where X = {Xt|t ∈ V (T )} is a family of subsets of V ,called bags, and T is a tree, such that: (1)⋃t∈V (T )Xt = V ; (2) for any edgeuv ∈ E, there is t ∈ V (T ) such that Xt contains both u and v; and (3) forany vertex v ∈ V , the set {t ∈ V (T )|v ∈ Xt} induces a subtree of T .The width of (T,X ) is maxt∈V (T )|Xt|− 1 and its size is the order |V (T )| ofT . If T is a path, (T,X ) is called a path-decomposition of G. The treewidth(resp., pathwidth) of G, denoted by tw(G) (resp., pw(G)), is the minimumwidth over all tree-decompositions (resp., path-decompositions) of G.Tree-decompositions are the corner-stone of many dynamic programmingalgorithms for solving graph problems. For example, the famous Courcelle’sTheorem states that any problem expressible in MSOL can be solved in linear-time in the class of bounded treewidth graphs [5]. Another framework based ongraph decompositions is the bi-dimensionality theory that allowed the designof sub-exponential-time algorithms for many problems in the class of graphsexcluding some fixed graph as a minor (e.g., [6]). Given a tree-decompositionwith width w and size n, the time-complexity of most of such dynamic pro-gramming algorithms can often be expressed as O(2wn) or O(2w logwn). There-fore, the problem of computing tree-decompositions with small width hasdrawn much attention in the last decades.The above-mentioned algorithms have mainly theoretical interest becausetheir time-complexity depends exponentially on the treewidth and, on theother hand, no practical algorithms are known to compute a good tree-de-composition for graphs with treewidth at least 5. However, for small (≤ 4)treewidth graphs, practical algorithms exist to compute tree-decompositionswith minimum width [11,2,10]. Since the computation of tree-decompositionsis a challenging problem, we propose in this paper to study it from a newpoint of view. Namely, we aim at minimizing the number of bags of the tree-1 This work has been partially supported by ANR project Stint (ANR-13-BS02-0007), theassociated Inria team AlDyNet, the project ECOS-Sud Chile and a grant from the ”Conseilrégional Provence Alpes-Côte d’Azur”.2 Email: bi.li@inria.fr3 Email: fatima-zahra.moataz@inria.fr4 Email: nicolas.nisse@inria.fr5 Email: karol.suchan@uai.cldecomposition when the width is bounded. This new perspective is interestingon its own and we hope it will allow to gain more insight into the difficulty ofdesigning practical algorithms for computing tree-decompositions.We consider the problem of computing tree-decompositions with mini-mum size. If the width is not constrained, then a trivial solution is a tree-decomposition of the graph with one bag (the full vertex-set). Hence, givena graph G and an integer k ≥ tw(G), we consider the problem of minimizingthe size of a tree-decomposition of G with width at most k.Related Work. The problem of computing “good” tree-decompositions hasbeen extensively studied. It is NP-hard to compute tree-decompositions withminimum width [1]. For any fixed k ≥ 1, Bodlaender designed an algorithmthat computes, in time O(kk3n), a tree-decomposition of width k of any n-node graph with treewidth at most k [3]. Recently, a single-exponential (in k)algorithm has been proposed that computes a tree-decomposition with width≤ 5k in the class of graphs of treewidth ≤ k [4].We are not aware of any work dealing with the computation of tree-decompositions with minimum size. In [7], Dereniowski et al. study theproblem of size-constrained path-decompositions. Given any k > 0 and anygraph G with pathwidth at most k, let lk(G) denote the smallest size (length)of a path-decomposition of G with width at most k. For any fixed k ≥ 4 (resp.k ≥ 5), computing lk is NP-complete in general graphs (resp., in the class ofconnected graphs) [7]. Moreover, computing lk can be solved in polynomial-time in the class of graphs with pathwidth at most k for any k ≤ 3. Thedual problem is also hard: for any fixed l ≥ 2, it is NP-complete in generalgraphs to compute the minimum width of a path-decomposition with length l[7]. Moreover, by the same techniques, the original proof can be extended toan analogous result on treewidth.Our results. For any tw > 0 and any graph G of treewidth at most tw, letsk(G) denote the minimum size of a tree-decomposition of G of width at mostk, k ≥ tw. We consider the complexity of computing sk for k ≥ 1. Table 1summarizes our results as well as the remaining open questions.s1 s2 s3 s4 sk, k = max{tw + 1, 5}Graphs of treewidth at most tw = 1 P (trivial) P P ? ?Graphs of treewidth at most tw = 2 - P ? ? ?Graphs of treewidth at most tw = 3 - - ? NP-hard ?Graphs of treewidth at most tw ≥ 4 - - - NP-hard NP-hardTable 12 Complexity results on the computation of skIn this section, we present some NP-hardness results and some polynomialalgorithms for the computation of sk for k ≥ 1. Due to lack of space, allproofs have been omitted and can be found in [8]. First we prove that:Theorem 2.1 For any fixed integer k ≥ 5, the problem of computing sk isNP-complete in the class of connected graphs with treewidth at most k − 1;furthermore, the problem of computing s4 is NP-complete in the class of planargraphs with treewidth at most 3.Our proof extends the one of [7] for size-constrained path-decompositions.In the rest of this extended abstract, we consider the computation of skfor k ≤ 3. We give polynomial-time algorithms in several graph classes. Letus first present the general approach that is used in what follows.General Approach. Let k ≥ 1 and G be a graph with tw(G) ≤ k. A subsetB ⊆ V (G) is a k-potential-leaf if there is a tree-decomposition (T,X ) withwidth at most k and size sk(G) such that B is a leaf bag of (T,X ) (i.e., thenode corresponding to bag B in T is a leaf). Abusing the notations, we willidentify the subset B with the subgraph it induces in G. Given a class ofgraphs C and a positive integer k, a set of graphs P is called a complete setof k-potential-leaves of C, if for any graph G ∈ C, there exists a graph H ∈ Psuch that G contains a k-potential leaf isomorphic to H.The key idea of our algorithms is to identify a complete set of potential-leaves. Then we proceed recursively: we find in G a k-potential-leaf H fromthe complete set, put it in a bag and combine in with the minimum sizetree-decomposition of G \H. The next lemmas formalize this idea.Given a graph G = (V,E) and a subset S ⊆ V , we denote by GS the graphobtained from G by adding the minimum number of edges to G such that Sinduces a clique in GS.Lemma 2.2 Let k ≥ 1 and G = (V,E) be a graph with tw(G) ≤ k. LetB ⊆ V be a k-potential-leaf of G. Let S ⊂ B be the set of vertices of B thathave a neighbor in V \B. Then sk(G) = sk(GS \ (B \ S)) + 1.This lemma implies the following corollary:Corollary 2.3 Let k ∈ N∗ and C be the class of graphs with treewidth at mostk. If there is a polynomial-time algorithm that, for any graph G ∈ C, computesa k-potential-leaf of G, then sk and a minimum size tree decomposition of widthat most k can be computed in polynomial-time in C.From Corollary 2.3, solving the Minimum Size Tree-Decomposition prob-lem in a class of graphs C reduces to the problem of finding a k-potential-leaffor any graph G ∈ C. This can be done in polynomial time if we characterizea finite complete set of k-potential-leaves of C. In the rest of this section, wepresent our results for k ∈ {2, 3}.Minimum Size Tree-Decompositions for k ∈ {2, 3}. We first charac-terize a complete set of 2-potential-leaves in the class of non-empty graphs(containing at least one edge) of treewidth at most 2 (see Fig. 1). Then, weidentify a complete set of 3-potential-leaves in the class of trees with at leastfour vertices and in the class of 2-connected outerplanar graphs (see Fig. 2).Note that the complete sets of potential-leaves in the figures are finite andeach potential-leaf in them can be found in polynomial time. This togetherwith Corollary 2.3 imply the following result:qpqbfcac affp bαFig. 1. Complete set of 2-potential-leaves of graphs of treewidth at most 2gghppfgfp'fgfp'fpf f 'f ''f(a) Treesaedcba dcbx yadcb(b) 2-connected outerplanar graphsFig. 2. Complete sets of 3-potential-leaves of trees and 2-connected outerplanarsTheorem 2.4 s2 and a minimum size tree decomposition can be computed inpolynomial-time in general graphs; s3 and a minimum size tree decompositionof width at most 3 can be computed in polynomial-time in the class of treesand 2-connected outerplanar graphs.3 ConclusionIn this extended abstract, we give preliminary results on the complexity ofminimizing the size of tree-decompositions with given width. Table 1 sum-marizes our results as well as the remaining open questions. We currentlyinvestigate the computation of s3 in the class of graphs with treewidth 2 andsk for k ≥ 3 in the class of trees. These cases are more intricate than thepolynomial cases we have considered. It seems that a global view of the graphneeds to be considered to decide whether a subgraph is a 3-potential leaf of thegraph; in addition, a k-potential leaf of a tree for k ≥ 5 can be disconnected(illustrating examples can be found in [8]).References[1] Arnborg, S., D. G. Corneil and A. Proskurowski, Complexity of findingembeddings in a k-tree, SIAM J. Algebraic Discrete Methods 8 (1987), pp. 277–284. URL http://dx.doi.org/10.1137/0608024[2] Arnborg, S. and A. Proskurowski, Characterization and recognition of partial3-trees, SIAM J. Algebraic Discrete Methods 7 (1986), pp. 305–314. URL http://dx.doi.org/10.1137/0607033[3] Bodlaender, H. L., A linear-time algorithm for finding tree-decompositions ofsmall treewidth, SIAM J. Comput. 25 (1996), pp. 1305–1317.[4] Bodlaender, H. L., P. G. Drange, M. S. Dregi, F. V. Fomin, D. Lokshtanovand M. 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