On the Treewidths of Graphs of Bounded
Degree
Yinglei Song*, Menghong Yu*
School of Electronics and Information Science Jiangsu University of Science and Technology Zhenjiang, Jiangsu 212003, China
*yingleisong@gmail.com(YS);ymhzj2691@163.com(MY)
Abstract
In this paper, we develop a new technique to study the treewidth of graphs with bounded de-gree. We show that the treewidth of a graphG = (V, E)with maximum vertex degreedis at mostð1−Ce−4:06 ffiffid
p
ÞjVjfor sufficiently large d, whereCis a constant.
Introduction
Tree decomposition is one of the most important concepts developed in graph theory in the past two decades. In a tree decomposition, graph vertices are grouped into vertex subsets, each of the vertex subsets is represented with a single node and all nodes are connected into a tree. Tree decomposition has provided original but profound insights into structural properties of graphs. For example, tree decomposition is the fundamental tool in the proof of the graph minor theorem [13,14,15,16,17,18]. On the other hand, tree decomposition also has impor-tant applications in algorithm design and complexity research. A generic dynamic program-ming framework has been available for solving many NP-hard problems on graphs using tree decomposition [2]. Based on this framework, important algorithmic and complexity results have been found for some graph theoretic optimization problems [3,9].
Fig. 1(b)shows a tree decomposition of the graph inFig. 1(a). Tree decomposition provides a new topological view in the structure of a graph and has been shown to be related to many deep properties related to graph minors [14,16]. These properties have played fundamental roles in the proof of the graph minor theorem. Moreover, tree decomposition also has impor-tant implications in algorithm design. For example, treewidth provides a structure parameter for a graph and many NP-hard graph optimization problems can be efficiently solved when the treewidth of the underlying graph is small [2]. More specifically, given a tree decomposition of a graph, we can easily identify subproblems for many NP-hard optimization problems and par-tial optimal solutions for these subproblems can be extended or combined with an exhaustive search performed only on vertices in a single tree node. As a result, given a tree decomposition, a dynamic programming approach can be employed to solve these optimization problems.
Treewidth is a structural parameter often used to evaluate a tree decomposition. The tree-width of a tree decomposition is determined by the maximum number of vertices contained in a single tree node and the treewidth of a graph is the minimum treewidth over all its tree
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OPEN ACCESS
Citation:Song Y, Yu M (2015) On the Treewidths of Graphs of Bounded Degree. PLoS ONE 10(4): e0120880. doi:10.1371/journal.pone.0120880
Academic Editor:M. Sohel Rahman, Bangladesh University of Engineering and Technology, BANGLADESH
Received:March 4, 2014
Accepted:February 10, 2015
Published:April 7, 2015
Copyright:© 2015 Song, Yu. This is an open access article distributed under the terms of theCreative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding:This work is fully supported by the New Faculty Start-up Funding at Jiangsu University of Science and Technology, China, under the funding number 635301202. The funder had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
decompositions. Based on the aforementioned dynamic programming framework, some
NP-hard optimization problems including MAXIMUMINDEPENDENTSETand MINIMUMD
OMINAT-INGSETcan be solved in timeO(f(t)jVj), given a tree decomposition of treewidthtfor the
un-derlying graphG= (V,E) [2]. However, sincef(t) is in general an exponential function oft, the treewidth of the available tree decomposition often determines the computational efficiency of this dynamic programming method.
A well known example is that, a maximum independent set in a graphG= (V,E) can be found in timeO(2kjVj) based on a tree decomposition of treewidthkforG[2]. A more generic result states that, any graph optimization problem that can be formulated with Monadic Sec-ond Order (MSO) logic can be solved in polynomial time when a tree decomposition of bound-ed treewidth is available for the underlying graph [8]. Treewidth is often important for the computational efficiency of these algorithms in the sense that the most computationally inten-sive part in such an algorithm arises from an exponential function of the treewidth.
Computing the treewidth of a graph is an NP-hard problem [1]. Due to the difficulty of finding the treewidth of a graph, a few efficient algorithms and methods have been developed to estimate its upper bounds and lower bounds [4,5,6,12]. For some graph families, an upper bound for treewidth can be estimated from other structural parameters with a simple formula.
Fig 1. An example of graph tree decomposition.(a) a graph; (b) a tree decomposition of the graph in (a).
For example, the treewidth of a planar graph is bounded from above by 3l+ 1 if the diameter of the graph isl[3,9]. The treewidth of a graph withmedges has been shown to be less thanm/5 [10]. In [7], an upper bound for treewidths are obtained fork-chordal graphs, wherekis a con-stant. An improvement of this bound is recently provided in [11]. However, due to the difficul-ty of relating treewidth to other structural parameters, new tools are needed to obtain similar results for some graph families, such as graphs with bounded degree.
In this paper, we consider graphs of bounded degree and develop a nontrivial asymptotic upper bound for the treewidths of such graphs. Based on a new technique,αtree backbone, we
show that the treewidth of a graph with maximum vertex degreedis asymptotically bounded
from above byð1 Ce 4:06pdffiffiÞjVjfor large enoughd, whereCis a constant. The result shows
that the technique ofαtree backbone can be effectively used to provide new insights into the structural properties of graphs of bounded degree.
Preliminaries
The graphs in this paper are undirected graphs without loops. For a given graphG= (V,E) and a vertexv2V,N(v) is the set of vertices that are connected tovby an edge inGandjN(v)jis thedegreeofv.N[v] denotes {v}[N(v). Thedegreeof a graph is the maximum degree of all its vertices. To simplify the notation, we useG−vto represent the graph obtained by removingv and all the edges incident tovfromG. For a vertex subsetU, we useG−Uto denote the graph obtained fromGby removing all vertices inUand the edges incident to them fromG. For a subsetUV,N(U) denotes the set of all vertices that are not inubut are joined by an edge to at least one vertex inU;G[U] is the subgraph induced byUinG. We used(G) to denote the size of the minimum independent dominating set in graphG. Apathin a graph is a sequence of verticesv1,v2, ,vlsuch that there is a graph edge betweenviandvi+1(1i<l). We useP = (v1,v2, ,vl) to represent a pathPof lengthl. Acycleof lengthlis a sequence of verticesv1,
v2, ,vlsuch that there is a graph edge betweenviandv(i+ 1)mod l.
Definition 1[13] LetG= (V,E) be a graph, whereVis the set of vertices inG,Edenotes the set of edges inG. The pair (T,X) is atree decompositionof graphGif it satisfies the following conditions:
1. T= (I,F) defines a tree, the sets of vertices and edges inTareIandFrespectively,
2. X= {Xiji2I,XiV}, and8u2V,9i2Isuch thatu2Xi,
3. 8(u,v)2E,9i2Isuch thatu2Xiandv2Xi,
4. 8i,j,k2I, ifkis on the path that connectsiandjin treeT, thenXi\XjXk.
Thetreewidth of the tree decomposition(T,X) is defined as maxi2IjXij−1. Thetreewidth of the graph Gis the minimum treewidth over all possible tree decompositions ofG.
Definition 2LetG= (V,E) be a graph, a vertex subsetFVis afeedback vertex setif graph G−Fis acyclic. Theminimum feedback vertex setin aGis the feedback vertex set with the min-imum number of vertices.
An Asymptotic Upper Bound for TreeWidth
Based on Lemma 3, an upper bound of the treewidth of a graph can be obtained from the size of a feedback vertex set in the graph. We focus on finding an upper bound for the mini-mum size of the feedback vertex set in a graph of degree bounded byd.
Definition 3For a graphG= (V,E) of degreed, anαtree backbone(0<α<1) of the graph is a subgraphTofGthat satisfies the following:
1. Tis a forest that contains a number of trees;
2. for each tree inTthat contains more than 2 internal nodes, each internal node is of degreed;
3. for any vertexvof degreed, ifv2=T, at leastαdvertices inN(v) are nodes of the trees inT.
For a graph with maximum vertex degreed, anαtree backbone is a maximal forest such
that each internal node in the forest is a vertex of degreedinGand has at least (1−α)dchild nodes in the tree. We prove in the following lemma that we are able to construct anαtree back-bone in polynomial time.
Lemma 2For a graphG= (V,E) of degreed, given a positive numberα<1, there exists an algorithm that can find anαtree backboneTin polynomial time.
PROOF. We show thatTcan be constructed by a greedy algorithm. Initially, we mark all
verti-ces inGto be unselected. We arbitrarily choose an unselected vertexvof degreedand markv and all vertices inN(v) as selected. We then proceed to vertices inN(v). For each vertexu2N (v), all of the unselected neighbors ofuare marked as selected if it is of degreedand the num-ber of its unselected neighbors isd−1. We recursively apply this procedure until we cannot mark more vertices as selected. We then include all selected vertices inT. For each remaining unselected vertex of degreed, we simply check the number of its neighbors that have been se-lected. If this number is smaller thanαd, we include the vertex and its unselected neighbors in Tand mark them as selected. It is not difficult to verify that this greedy algorithm can find anα
tree backboneTin polynomial time.
For a given positiveα<1 and a graphG= (V,E) of degree at mostd, based on anαtree backboneTof the graph, any vertex of degreedthat is not inThas at leastαdneighbors that are leaves of the trees inT. The following lemma considers only the vertices that are of degreed and not inT.
Theorem 1Given a graphG= (V,E) of maximum degreedand a positive numberα<1, if Tis anαtree backbone inG, there exists a vertex subsetSthat satisfies the following:
1. A connected component inG−Sis either a tree or a graph of degree bounded byd−1;
2. jSj ð 2
ð1 aÞdþ
2d 1þpffiffid
aðpffiffid þ1ÞdÞjVj.
PROOF. For a givenαtree backboneTthat containsmvertices, we consider the set of the
in-ternal nodes that are connected to a leaf in a tree inT. We denote such a set withI. We assume jIj=sand denote the nodes inIwithv1,v2, ,vs. In addition, we used1,d2, ,dsto denote
the degree ofv1,v2, ,vsinT. It is not difficult to see that the following inequality holds.
Xs
i¼1
di2ðm 1Þ<2m ð1Þ
The inequality holds since each edge is counted at most twice in the summation and there are at mostm−1 edges inT. On the other hand, since the degree of each node inIis at least (1−
α)d, we also have the following inequality.
Xs
i¼1
Therefore, we can obtain
s< 2m
ð1 aÞd ð3Þ
We thus havejIj< 2m
ð1 aÞd. We now consider the vertices that are of degreedand not inT.
We assume that such vertices form a vertex setV1and their neighbors inTform vertex setV2. The bipartite subgraph formed betweenV1andV2isH= (V1[V2,F), such that for verticesu 2V1andv2V2, (u,v)2Fif and only if (u,v)2E. The degree of vertices inV1inHis at least
αd. Note that we can assume the degree of each vertex inV2inHis at mostd−1, since each such vertex is connected to at least one vertex inTby an edge and its degree is at mostdinG.
We now show thatHcontains a dominating setDof size no larger than2d 1þpffiffid
aðpffiffidþ1Þdm. To see
this, we apply the following operations to vertices inV2. Ifv2V2and it’s degree inHis larger than a given positive numberc, we simply includevinDand remove its neighbors inV1from H. We repeat this until no remaining vertex inV2have degree more thancinH, we then in-clude all the remaining vertices inV1inD.
It is not difficult to see that the number of vertices that are inV2and included inDis at mostjV1j/(c+ 1), for the remaining vertices inV1since their degree is at leastαdin graphH, we haveαdjV0j cjV2j, whereV0is the set of remaining vertices inV1. Combining the two we can obtain that the size of this dominating set is at mostjV1j
cþ1þ
cjV2j
ad. On the other hand, since we
haveαdjV1j (d−1)jV2j, we getjDj
d 1þcðcþ1Þ
ðcþ1Þad jV2j. Now if we letc¼
ffiffiffi d p
and we obtain a
dominating set with its size bounded by2d 1þpffiffid
aðpffiffidþ1ÞdjV2jfrom above.
We now consider the setS=I[D, it is not difficult to see that after removing all vertices in S, each connected component in the resulting graphG−Sis either a tree or a graph of degree bounded byd−1. and we havejSj ð
2 ð1 aÞdþ
ð2d 1þpffiffidÞ
aðpffiffidþ1ÞdÞjVj. The theorem has been proved.
Based on Theorem 1, we are able to compute an asymptotic upper bound for the size of the
minimum feedback vertex set in a graphGwith maximum vertex degreed, wheredis an
inte-ger that is sufficiently large. Specifically, Theorem 1 states that we are able to remove a fraction of approximately2=pffiffiffidof the vertices from such a graph such that each connected component
in the resulting graph is either a tree or a graph with its maximum degree bounded byd−1. We can recursively apply this procedure to each connected component that is not a tree and we show that this procedure can lead to an upper bound estimate for the size of the minimum feedback vertex set.
Lemma 3Given a graphG= (V,E) of maximum degreed, there exists an independent ver-tex subsetSVsuch thatjSj jVj/2 andG−Sis a graph of maximum degreed−1.
PROOF. To obtain such anS, we can arbitrarily choose a vertexuof degreedand include it in S. We then removeuand the edges incident to it fromGand recursively apply the same proce-dure on the resulting graph until no such vertices can be found.Sinduces an independent set inGand we havejEj djSj. On the other hand,jEj djVj/2. This leads tojSj jVj/2.
Definition 4Given a graph familyFand an algorithmAthat can compute a feedback vertex set in a graph inF, theupper bound function u(F,A) is defined asmaxG2Ff
BðGÞ
jVjg, whereG= (V, E) is a graph inFandB(G) is the feedback vertex set computed byAinG.
does not contain a connected component that is not a tree. Otherwise, each component that is not a tree inG−Sis recursively processed. In the rest of the paper, we useA1to denote the
al-gorithm developed based on Theorem 1 andA2for the algorithm developed based on
Lemma 3.
Theorem 2For a given graphG= (V,E) of maximum degreed, there exists a positive con-stantCsuch that for a sufficiently larged,Gcontains a feed back vertex set with size at most ð1 Ce 4:06pffiffid
ÞjVj. Such a feedback vertex set can be found in polynomial time.
PROOF. From Theorem 1, we can letα= 0.99 and there exists an integerD0such that when d>D0, we can find a vertex subsetSVthat satisfies the property that each connected com-ponent ofG−Sis either a tree or a graph with maximum degree ofd−1 and
jSj 2:021jVj=pffiffiffid. An additional requirement for selectingD0is specified later in the proof.
Based onD0, the following algorithmA3can compute a feedback vertex set in a graph G= (V,E).
1. For each connected componentMthat is not a tree inGand does not contain a vertex of de-gree larger thanD0, applyA2toMto get a feedback vertex setFinMand include all vertices inFinto the feedback vertex set;
2. the algorithm returns ifGdoes not contain a connected component that is not a tree and contains at least one vertex of degree larger thanD0. Otherwise it continues to step 3;
3. for each connected componentEthat is not a tree inGand contains at least one vertex of degree larger thanD0, find a subsetSinEas described in the proof of Theorem 1, include all vertices inSinto the feedback vertex set and removeSfromE;
4. set the resulting graph to beG0and recursively apply the algorithm toG0;
From Definition 4, a graphG= (V,E) with maximum degreedhas a feedback vertex set of size at mostu(L(d),A3)jVj. From the above description ofA3, we obtainu(L(D0),A3) =u(L (D0),A1).A3thus returns a feedback vertex set of size at mostu(L(D0),A1)jVjin a graphG= (V,E) with maximum degreeD0. From Lemma 3, we immediately obtain thatu(L(d),A1) must satisfyu(L(d),A1)(u(L(d−1),A1) + 1)/2. If we consider the case whered= 2, we find that uðLð2Þ;A
1Þ 1
2. For a finite integerD0, it is obvious thatu(L(D0),A1) is a positive constant
strictly less than 1. Sinceu(L(D0),A3) =u(L(D0),A1),u(L(D0),A3) is also a positive constant strictly less than 1.
We assumeA3returns a feedback vertex set of sizeU(L(d),A3)jV1jin graphG1= (V1,E1)2 L(d). From Theorem 1, it is clear that whend>D0, we can find a vertex subsetS1V1such
that each connected component ofG1−S1is either a tree or a graph with maximum degree of
d−1 andjS1j 2:021jV1j=
ffiffiffi d p
. We useF(G1−S1) to denote the feedback vertex set obtained byA3onG1−S1. Since each connected component ofG1−S1is either a tree or a graph inL(d
−1), it is not difficult to see that the following inequality holds
jFðG1 S1Þj uðLðd 1Þ;A3ÞðjV1j jS1jÞ ð4Þ
On the other hand, sinceA3returnsS1[F(G1−S1) as a feedback vertex set inG1, we have
uðLðdÞ;A3ÞjV1j ¼ jFðG1 S1Þj þ jS1j ð5Þ
uðLðd 1Þ;A
3ÞðjV1j jS1jÞ þ jS1j ð6Þ
¼ ð1 uðLðd 1Þ;A
Sinceu(L(d−1),A3)1 andjS1j 2:021jV1j=
ffiffiffi d p
, we can immediately obtain the following recursion relation foru(L(d),A3) whend>D0.
uðLðdÞ;A3Þ
2:021
ffiffiffi d
p ð1 uðLðd 1Þ;A
3ÞÞ þuðLðd 1Þ;A3Þ ð8Þ
Since the following relation holds for a sufficiently small positive numberx
1 xe 1:004x
ð9Þ
it is clear that we can select sufficiently largeD0such that whend>D0, we have
1 2ffiffiffiffiffiffiffiffiffiffiffi:021 dþ1
p e p2ffiffiffiffiffid:03þ1 ð10Þ
On the other hand, we have
2:03
ffiffiffiffiffiffiffiffiffiffiffi dþ1
p < ffiffiffiffiffiffiffiffiffiffiffi4:06
dþ1
p
þ ffiffiffi d
p ð11Þ
¼ 4:06ð ffiffiffiffiffiffiffiffiffiffiffi dþ1
p ffiffiffi
d p
Þ ð12Þ
We can thus obtain
1
2:021
ffiffiffiffiffiffiffiffiffiffiffi dþ1
p e 4:06ðpffiffiffiffiffiffidþ1 pffiffidÞ
ð13Þ
We can choose an appropriate constantCthat satisfiesuðLðD0Þ;A3Þ 1 Ce 4:06pffiffiffiffiD0
. We next show thatuðLðdÞ;A3Þ ð1 Ce
4:06pffiffidÞwhendD0. Based on the principle of
induc-tion, whend=D0the relation holds due to the selection of constantC. Now, we assume that the relation holds whend=l, based on the recursion relation, we have
uðLðlþ1Þ;A
3Þ ð1 uðLðlÞ;A3ÞÞð1 e
4:06ðpffiffiffiffiffilþ1 plffiÞÞ þuðLðlÞ;A
3Þ ð14Þ
1 e 4:06ðpffiffiffiffiffilþ1 pffilÞ
ð1 uðLðlÞ;A
3ÞÞ ð15Þ
1 Ce 4:06pffiffiffiffiffilþ1
ð16Þ
Based on the principle of induction, we knowu(L(d),A3) is at most1 Ce 4:06pffiffid
whendD0.
In addition, from Lemma 2,A3needs polynomial time tofind such a feedback vertex set. Here, we need to point out that, based on Lemma 3, we can also construct a recursive relation forA2. However, this recursion can only lead to an upper bound estimation of (1−1/2O(d))jVj. This trivial asymptotic upper bound now has been significantly improved using the technique ofαtree backbone. Based on the relation between the treewidth and the size of a minimum feedback vertex set, we also obtain an asymptotic upper bound for the tree-width of a graph with bounded maximum vertex degree.
Corollary 1For a given graphG= (V,E) with maximum vertex degreed, there exists a posi-tive constantCsuch that for a large enoughd, the treewidth of the graph is bounded from above byð1 Ce 4:06pffiffid
Theorem 2 also provides an asymptotic upper bound for the treewidths of sparse graphs. Specifically, for a graphG= (V,E), wherejEj ΔjVjandΔis a positive constant independent
ofG, we have the following theorem.
Theorem 3Given a graphG= (V,E), wherejEj ΔjVjandΔis a positive constant
indepen-dent ofG, there exists a positive constantC0such that for a large enoughΔ, the treewidth ofG
is bounded from above byð1 C0e 8:12pffiffiD
ÞjVj.
PROOF. First, sincejEj ΔjVj, the number of vertices with degree larger than 4Δis at most
jVj/2. We assume the number of such vertices isβjVjand we have 0β1/2. Now, we re-move all these vertices and edges incident to them fromGand the resulting graphG0is a graph with maximum vertex degree 4Δ. Apply Theorem 2 to graphG0, for large enoughΔ, we can
ob-tain a tree decompositionT0forG0with its treewidth at mostð1 Ce 8:12pffiffiD
ÞjV0j, whereCis a
positive constant. We can thus obtain a tree decompositionTforGby including the removed vertices in each tree node ofT0. The tree width ofTis at most
bjVj þ ð1 Ce 8:12pffiffiD
Þð1 bÞjVj. Sinceβ1/2, the treewidth ofTis bounded from above by
ð1 C
2e 8:12pffiffiD
ÞjVj.
Conclusions
In this paper, we provided an original perspective on the structure of a graph with bounded de-gree. We develop a new technique,αtree backbone, to analyze the treewidth for graphs of bounded degree. Our analysis leads to a nontrivial asymptotic upper bound for the treewidth of such graphs.
We have seen in the proof of Theorem 1 that the size of the dominating setDin the bipartite graphHis crucial for our analysis and improvements made on its size can lead to further im-proved bounds for treewidth. In particular, we conjecture that the treewidth of a graphG= (V,
E) with maximum vertex degreedis in fact asymptotically bounded byð1 C
dcÞjVj, whereC andcare constants. However, the technique based onαtree backbone may not be sufficient to obtain such an upper bound. Our future work may focus on providing a proof or counterexam-ple for this conjecture.
Acknowledgments
The authors are grateful for the constructive comments from the anonymous reviewer on an earlier version of the paper.
Author Contributions
Conceived and designed the experiments: YS. Analyzed the data: YS. Contributed reagents/ma-terials/analysis tools: MY. Wrote the paper: YS MY.
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