can be computed inO(kk(T,X)k)time, when the tree decomposition(T,X)is provided as input.
1.2.6 Further Remarks
Most results from Sections1.2.1-1.2.5do not only hold for bisections but also for cuts (B, W) where the size of the setB is specified as input. When presenting the proof of such a result, it is stated in its general form.
Here, only bisections andk-sections in unweighted graphs are considered. In many applications, the vertices and the edges of the considered graph have weights. Any upper bound for the width of a minimum bisection in a graph without edge weights can be adjusted for graphs with edge weights by multiplying it with the maximum edge weight. For vertex-weights, the situation is more difficult. In a graphG= (V, E) with vertex weights, deciding whether V allows a partition (B, W) such that the sum of the weights inB equals the sum of the weights in W is equivalent to solving the NP-complete Subset Sum Problem, see Problem MP9 in [GJ79]. In [Ham16], a bisection in a graph Gwith vertex weights is defined as a partition (B, W) of V(G) such that |g(B)−g(W)| ≤ gmax, where g(B) and g(W) denote the sum of the weights inB andW, respectively, andgmax is the maximum vertex weight inG. Then, a version of Theorem 1.1 for trees with vertex weights and a corresponding algorithm is derived. Furthermore, in [Ham16], also Lemma1.12and Theorem1.15are generalized for trees with vertex weights.
1.3 Organization of the Thesis
After presenting some preliminaries in the next chapter, the proofs of the results summarized in Section1.2 are presented in the following order.
Chapter 3 focuses on planar graphs. First, the method for constructing bisections in graphs via separators that was sketched in Section1.2.3is made precise and then used to derive Theorem1.10and Theorem1.11. Furthermore, Theorem1.4is proved. Afterwards, Chapter3investigates bounded-degree planar graphs with large minimum bisection width. Grid-homogeneous graphs are introduced in detail and Theorem 1.7 is proved. Moreover, algorithmic aspects related to grid-homogeneous graphs as mentioned in Section1.2.2are discussed.
Chapter 4 studies approximate cuts in trees and tree-like graphs. In particular, Lemma 1.12 and Lemma1.13are proved here. Furthermore, Corollary1.14is derived in Chapter4.
Chapter 5 concentrates on bisections in trees and tree-like graphs. First, the methods used throughout the chapter are introduced slowly by studying trees T on n vertices with diam(T) ≥ 14n. Second, Theorem1.1is proved and an algorithm computing a bisection whose width is at most as large as the bound in Theorem1.1is described. Afterward, the analysis is tightened to yield Theorem 1.8. Third, the methods are generalized to tree-like graphs and Theorem1.9is proved. Observe that Theorem1.3 does not need to be proved separately as Theorem1.9is stronger.
Chapter 6 focuses onk-sections. To begin with, it is shown that simple, recursive approaches using Theorem1.1to construct ak-section in a tree can yield ak-section of width much larger than promised by Theorem1.15. Then, the proof of Theorem1.15is presented and the analysis is tightened as described in Section 1.2.5. Furthermore, the methods are generalized to tree-like graphs and Theorem 1.16is derived.
Figure1.4visualizes connections between the results proved in this thesis. All results are joint work with Cristina G. Fernandes and Anusch Taraz. Parts of the results in Chapter 3and parts of the results for trees in Chapter5 have been published in the proceedings of EuroComb 2013 [FST13]. Some results on the Minimum k-Section Problem in trees from Chapter6have appeared in the proceedings of LAGOS 2015 [FST15b]. The extensions to tree-like graphs in Chapter5and Chapter6have been published in the proceedings of EuroComb 2015 [FST15a]. The journal version of the results from Chapter4and Chapter5has been submitted [FSTa]. Journal versions concerning the results of Chapter3and Chapter6 are in preparation [FSTb;FSTc].
Lemma1.12
Theorem1.1
Corollary1.2
Theorem1.8
Theorem1.15
Lemma1.13
Theorem1.3 Corollary1.14
Theorem1.9
Theorem1.11
Theorem1.16
Theorem1.10
Theorem1.4
Corollary 1.5
Theorem1.7
Figure 1.4:Visualization of the connections between the results proved in this thesis. Green color indicates that the result is for trees, blue for tree-like graphs, and red for planar graphs. A continuous arrow fromAtoBmeans thatAis used as a tool in the proof ofB. A dotted arrow fromAtoB means thatBgeneralizes the statement ofAto tree-like graphs.
Preliminaries and Notation
This chapter introduces the basic notation used throughout the thesis and states some general knowledge about graphs and algorithms. First, Section 2.1 introduces the basic notation concerning graphs and cuts. In particular, the notation for bisections andk-sections is introduced in Section2.1. Afterward, Section2.2 states some facts about graphs. Some basic facts stated there will be used in the following chapters without further mentioning them and others are stated to be able to refer to them later. Tree decompositions are introduced in Section 2.3, where also the properties (T1), (T2), (T3), and (T3’) are stated, which are referred to by these names throughout the entire thesis. Preliminaries concerning algorithms are introduced in Section2.4, which first discusses algorithms for graphs and then algorithms receiving a tree decomposition as input.
2.1 Basic Definitions
Before starting with the notation involving graphs, a few basic definitions are presented for the sake of completeness. LetN={1,2,3, . . .} be the set ofnatural numbers and defineN0=N∪ {0}. Forn∈N, let [n] ={1,2, . . . , n}. Sometimes, it will be useful to write [0], which is defined as the empty set. As usual, for a realx, the largest integeriwithi≤xis denoted bybxc, and the smallest integeriwith i≥x is denoted bydxe.
Graphs
In this thesis, all graphs are finite, undirected, and do neither have vertex nor edge weights. So a graphG= (V, E) consists of a finite vertex set and an edge setE⊆ V2:={{v, w}: v∈V, w∈V, v6=w}. Except for a few occurrences in iterative procedures, the vertex set of a graph is always assumed to be non-empty. For a graphG, denote byV(G) itsvertex set and byE(G) itsedge set.
Consider a graphG= (V, E). An edge e∈E and a vertexv∈V are calledincidentifv∈e. For two verticesv, w∈V, the vertexwis aneighbor ofv if{v, w} ∈E. Ifv is a neighbor ofw, we also say thatv andwareadjacent. The set of neighbors ofv is called theneighborhood ofv and is denoted byNG(v).
For a vertexv∈V, let degG(v) denote itsdegree, which is defined as the number of edges inE that are incident tov or, equivalently, the number of neighbors ofv. Sometimes, when it is clear from the context to which graph the degree refers, then deg(v) is used instead of degG(v). A vertexv∈V with degG(v) = 0
is called anisolated vertex and a vertexv∈V with degG(v) = 1 is called aleaf of G. Themaximum and minimum degreeofGare defined as ∆(G) := max{degG(v): v∈V}andδ(G) := min{degG(v): v∈V}, respectively. If there is ad∈Nsuch that the graphGsatisfies degG(v) =dfor allv∈V, then Gis called d-regular.
Often, subgraphs, that are formed by removing vertices or edges from another graph, will be studied.
Consider again a graphG= (V, E). A graphH is asubgraphofGifH satisfiesV(H)⊆V andE(H)⊆E andH itself is a graph. IfH is a subgraph ofG, then we writeH ⊆G. Induced subgraphs are subgraphs that, for a certain vertex set, contain all possible edges. More precisely, for a set∅ 6=W ⊆V, letG[W] be the graph with the vertex setW and the edge setE∩ {{v, w}: v, w∈W, v6=w}. Then,G[W] is a subgraph ofGand is called thesubgraph ofGinduced byW. Furthermore, for a setW ⊆V withW 6=V, the graph obtained fromGby removing all vertices inW as well as their incident edges is denoted byG−W. Observe thatG−W is an induced subgraph of G, namely G−W =G[V \W]. For setsW ={v}, we also write G−v instead of G− {v}. Similarly, for removing edges, consider a setF ⊆E and denote byG−F the graph obtained fromGby removing each edge inF and, ifF ={e}, thenG−eis used to abbreviateG− {e}. Note that, for an edgee={v, w}, the notationG−ecan be interpreted in two ways.
On the one hand, it can refer to removing the edgeeand, on the other hand, it can refer to removing the verticesv andw. Throughout the thesis, the correct interpretation will be stated explicitly or will be clear from the context.
Cuts, Bisections, andk-Sections
LetG= (V, E) be a graph and letn:=|V|. Acut inGis a partition ofV into several setsB1, . . . , Bk
with k≥2 and is denoted by (B1, B2, . . . , Bk). Here, the sets B` of a cut (B1, . . . , Bk) are allowed to be empty. For example, (V,∅) is a cut in G. Fix an integer k ≥2 and a cut (B1, . . . , Bk) in G. An edge{v, w} ∈Eiscut by (B1, . . . , Bk) if there are two distinct indices`, `0∈[k] withv∈B`andw∈B`0. The set of edges inE that are cut by (B1, . . . , Bk) is denoted byEG(B1, . . . , Bk). Moreover, thewidth of (B1, . . . , Bk) is defined as the number of edges cut by (B1, . . . , Bk) and is denoted byeG(B1, . . . , Bk).
So,eG(B1, . . . , Bk) :=|EG(B1, . . . , Bk)|. If the graphGis clear from the context, thenEG(B1, . . . , Bk) andeG(B1, . . . , Bk) are also abbreviated toE(B1, . . . , Bk) ande(B1, . . . , Bk), respectively. Ifk= 2, then usually (B, W) is used instead of (B1, B2) to denote a cut into two pieces and the sets B andW are referred to as theblack set and thewhite set, respectively. In the following, unless indicated otherwise, when a figure displays some cut (B0, W0) in a graphG0, then the vertices inB0 are colored black, the vertices inW0 are colored white, and each edge of G0 that is cut by (B0, W0) is colored red.
Let k≥2 be an integer and letG= (V, E) be a graph. A k-sectioninGis a cut (B1, . . . , Bk) in G with||B`|−|B`0|| ≤1 for all`, `0 ∈[k], i. e., ifndenotes the number of vertices ofG, thenn
k
≤ |B`| ≤n k
for all`∈[k]. Theminimum k-section width ofGis defined as
MinSeck(G) := min{eG(B1, . . . , Bk): (B1, . . . , Bk) is a k-section inG}.
Ak-section (B1, . . . , Bk) inGwith eG(B1, . . . , Bk) = MinSeck(G) is called aminimumk-sectioninG. A bisectioninGis a 2-section inG, i. e., a cut (B, W) inGthat satisfies|B|=|W|if the number of vertices ofGis even, and||B| − |W||= 1 if the number of vertices ofGis odd. Similarly, the minimum bisection width of Gis defined as MinBis(G) := MinSec2(G) and a minimum 2-section inGis called aminimum bisection inG.
Consider a graphG= (V, E) and letn:=|V|. Form∈[n], anm-cutinGis a cut (B, W) with|B|=m.
So, ann
2-cut inGis a bisection inG. Approximate cuts relax this size constraint in the following way.
Form∈[n], the cut (B, W) is called asimple approximate m-cut inGif 12m <|B| ≤m. Furthermore, for 0≤c <1 andm∈[n], a cut (B, W) inGis ac-approximatem-cut inGifcm≤ |B| ≤mand (B, W)
is astrict c-approximatem-cutinGifcm <|B| ≤m. Whenever the precise values of c andmdo not matter, we omit them and use the termapproximate cut. Sometimes, the termexact m-cutis used to refer to anm-cut in order to distinguish it from an approximatem-cut. The term exact cutis used to refer to an exactm-cut, when no specific value formis given, but the size of the black set is specified by an input parameter.
Special Graphs and Paths
For an integern≥2, thecomplete graphonnvertices is denoted byKn and defined as the graph with vertex set [n] and edge set [n]2. For two integersn≥1 andm≥1, thecomplete bipartite graph with partition classes of sizenandmis denoted byKn,mand is defined as the graph with vertex set [n+m] and edge set{{v, w}: v∈[n], w∈[n+m]\[n]}. For an integern≥2, thestar onnvertices is defined as the complete bipartite graphK1,n−1. Ifn≥3, then the unique vertexv of the starK1,n−1with deg(v)≥2 is called the center vertex of the star. LetP0:= ({0},∅) and, for n∈N, let Pn be the graph with vertex set [n]∪ {0} where two distinct verticesv and ware adjacent if and only if |v−w| ≤ 1. Forn∈ N0, the graphPn is called thepath of length n. Observe thatPn containsn+ 1 vertices and nedges. For an integern≥3, thecycleof length nis denoted byCn and defined as the graph obtained fromPn by removing the vertex 0 and adding the edge{1, n}.
Two graphs G = (V, E) and G0 = (V0, E0) are isomorphic if there is a bijection f : V → V0 such that {v, w} ∈E if and only if{f(v), f(w)} ∈E0. Fix a graph G= (V, E) and letv, w ∈V. Then, G contains av,w-pathif there is a sequence (u0, u1, . . . , u`) withu0=v,u`=w,uh∈V for allh∈[`]∪ {0} and{uh−1, uh} ∈Efor allh∈[`] such thatuh6=uh0for all distincth, h0 ∈[`]∪{0}. LetVP :={u0, . . . , u`} andEP :={{uh−1, uh}: h∈[`]}. Then, the subgraphP = (VP, EP) ofGis called apathof length`inG and is denoted by (u0, u1, . . . , u`). The verticesu0=v andu`=w are called theendsof P. Observe thatP is isomorphic toP`and thatvandware only leaves ofP if`≥1. IfP is av,w-path inG, then we say thatP joinsv tow. Forh, h0∈[`]∪ {0}with h≤h0, the path (uh, . . . , uh0) is called thesubpathofP that joinsuhtouh0. Similarly, for`≥3, (u1, . . . , u`) is acycle of length`inGwhenuh∈V for allh∈[`], {uh, uh+1} ∈E for all h∈[`−1], and {u`, u1} ∈E as well as that uh6=uh0 for all distincth, h0 ∈[`].
Observe that, in this case,Gcontains a subgraph with vertex set{u1, . . . , u`}that is isomorphic to the cycleC`. A vertexv∈V is on the pathP = (u0, . . . , u`) if there is anh∈[`]∪ {0}withv=uh. Similarly, an edgee∈E is on the pathP = (u0, . . . , u`) if there is an h∈[`] with{uh−1, uh}=e. Two paths P andP0 inGwithV(P)∩V(P0) =∅ are calledvertex-disjoint and, ifE(P)∩E(P0) =∅, then P andP0 are callededge-disjoint. Moreover, two pathsP = (u0, u1, . . . , u`) andP0= (u00, u01. . . , u0`0) are internally disjointifuh6∈V(P0) for allh∈[`−1] anduh0 6∈V(P) for allh0 ∈[`0−1]. Observe that two internally disjoint paths may have one or two common ends. A walk is a concept that is less strict than a path. A sequence (u0, u1, . . . , u`) of vertices is called awalkinG, ifuh∈V for allh∈[`]∪ {0}and{uh−1, uh} ∈E for all h ∈ [`]. Observe that every path is a walk but not vice versa as a walk may reuse a vertex.
Similarly as for paths,u,v-walks, vertex-disjoint walks, edge-disjoint walks, and internally disjoint walks are defined. Moreover, ifGcontains au,v-walk, thenGalso contains au,v-path, as each walk that is not a path can be shortened to become a path. For two pathsP = (u0, . . . , u`) andP0 = (u00, . . . , u0`0) with u` =u00, the walk obtained by glueing P and P0 together is the sequence (u0, . . . , u`, u01, . . . , u0`0).
Observe that (u0, . . . , u`, u01, . . . , u0`0) is not necessarily a path and, hence, the concept of walks is useful.
Miscellaneous
Let G = (V, E) be a graph. Then, G = (V, E) is connected if for all v, w ∈ V there is a v,w-path inG. A maximal connected subgraph of G, i. e., a subgraphH ⊆ Gsuch that there is no connected
subgraphH0⊆GwithH ⊆H0 andH 6=H0, is called acomponentof G. Moreover,Gis calledacyclic ifGdoes not contain a subgraph that is isomorphic toC` for all integers`≥3. Aforest is defined as an acyclic graph and atree is a connected and acyclic graph. Here, the symbolT is usually used to denote a graph that is a tree.
Roughly speaking, to subdivide an edgeemeans to insert a new vertex one, and to contract an edgee means to merge its vertices together. More precisely, consider a graph G= (V, E) and lete={u, v}
be an edge ofG. Subdividing emeans to removee fromG, to insert a new vertexw, and to insert the edges {u, w} and{v, w}. If G0 is a graph that is obtained from G by successively subdividing edges, thenG0 is called asubdivision ofG. Tocontract emeans to removeuandv fromGand to insert a new vertexwthat is adjacent to each vertex inNG(u)∪NG(v). IfG0 is isomorphic to a graph that is obtained from a subgraph ofGby successively contracting edges, thenGcontainsG0 as a minor. LetG0= (V0, E0) be a graph that is obtained fromGby successively contracting edges. Then one can partition V into setsMxwithx∈V0 such that the following properties are satisfied:
• For every x∈V0, the setMx is non-empty.
• For every x∈V0, the graphG[Mx] is connected.
• For every x, x0 ∈V, the graphG0 contains the edge{x, x0}if and only if Gcontains an edge{v, v0} withv∈Mx andv0∈Mx0.
So, when contracting all edges inG[Mx] for eachx∈V0 and calling the resulting vertexx, the graphG0 is obtained, see also Chapter 1.7 in [Die12].
Consider a graphG= (V, E). A pathP ⊆Gis called alongestpath in GifGcontains no pathP0 with|E(P0)|>|E(P)|. For two verticesv, w∈V, a path P⊆Gis called ashortestv,w-path inGifG contains nov,w-pathP0 with |E(P0)|<|E(P)|. Thedistance of two verticesv andwin a graphGis the length of a shortestv,w-path inGand is denoted by distG(x, y). For a connected graph, thediameter is defined as
diam(G) = max{dist(x, y): x, y∈V(G)}.
Observe that the diameter of a treeT is the length of a longest path in the tree as, forv, w∈V(T), every v,w-path inT is a shortestv,w-path.
The symbolRdenotes the set of real numbers and R>0denotes the set of positive, real numbers. The following asymptotic notation is used. Letf, g:R>0→R>0be two functions. We writef(x) =O(g(x)) if there are two constants c ∈ R>0 and x0 ∈ R>0 such that f(x) ≤cg(x) for all x ≥x0. Moreover, we write f(x) = Ω(g(x)) if there are two constantsc ∈R>0 andx0 ∈R>0 such thatf(x)≥cg(x) for all x ≥ x0, i. e., f(x) = Ω(g(x)) is equivalent to g(x) = O(f(x)). Finally, we write f(x) = Θ(g(x)) iff(x) =O(g(x)) andf(x) = Ω(g(x)).