Here, some basic knowledge about graphs is presented. In the following, most of these facts are used without further mentioning.
Proposition 2.1 (see Theorem 1.5.1 and Corollary 1.5.3 in [Die12]).
For every graphT = (V, E)the following statements are equivalent:
(i) T is a tree.
(ii) T is connected and|E|=|V| −1.
(iii) For everyv, w∈V, there is exactly onev,w-path inT.
In particular, it follows that every treeT = (V, E) satisfies|E|=O(|V|), which will be used often in the following. Consider a graphG= (V, E). As every edge in E contributes 1 to the degree of exactly two vertices,P
v∈V degG(v) = 2|E|. For trees and forests, the following corollary is obtained.
Corollary 2.2.
a) For every tree T = (V, E), the following holds: P
v∈V degT(v) = 2|V| −2. b) For every forestG= (V, E), the following holds: P
v∈V degG(v)≤2|V| −2.
Two Graphs and their Minimum Bisection Width
Here, two graphs are introduced, namely perfect ternary trees and grids. Moreover, bounds on their minimum bisection width are stated. To define the former one, the standard terminology for rooted trees is used, which is as follows. Arooted tree is a tree, where one vertex has been designated theroot. Consider a treeT with rootrand let xandy be two vertices in T. Then, y is called adescendant ofx ifxis on the uniquer,y-path inT. Note thatxis a descendant of itself. The parent of a vertexx6=r, denoted byp(x), is the neighbor ofxthat is on thex,r-path. Ify is the parent ofx, thenxis achild ofy.
In a rooted tree, aleaf is a vertex with no child. Observe that this differs slightly from the definition of a leaf in a graph asv is a leaf in the rooted tree ({v},∅) but not in the graph ({v},∅). Thesubtree rooted in xis the subgraph ofT that is induced by all descendants ofx. Leth, k be two integers with k≥2 andh≥0. A k-arytree is a rooted treeT = (V, E) with the property that every vertexv ∈V has at mostk children. A 2-ary tree is also called abinarytree and a 3-ary tree is also called aternary tree. A k-ary treeT = (V, E) is calledfull if every vertexv∈V has exactlykchildren or is a leaf ofT. A full, k-ary tree T, where each leaf of T has distancehto the root, is called a perfect k-ary tree of height h.
Clearly, a perfectk-ary tree of height 0 has 1 vertex, and a perfectk-ary tree of height 1 hask+ 1 vertices.
Using that a perfectk-ary tree of height his obtained by takingkvertex disjoint perfectk-ary trees of heighth−1, adding a new vertex rthat is the root, and joiningr to each root of the kperfectk-ary trees of heighth−1, the next proposition follows by induction.
Proposition 2.3.
For everyh∈N0 and every integerk≥2, a perfect k-ary tree of heighthhas k−11 kh+1−1
vertices.
In [Sch13], the following bounds on the minimum bisection width in perfect ternary trees are derived.
Theorem 2.4 (see Corollary 4.12 in [Sch13]).
For everyh∈N, every perfect ternary treeTh of height hsatisfies
h−log3(h) ≤ MinBis(Th) ≤ h−log3(h) + 3.
The second graph introduced in this paragraph is thegrid. For eachk∈N, the graph Gk= (Vk, Ek) defined by
Vk:={(i, j): i∈[k], j ∈[k]} and Ek:=
{(i, j),(i0, j0)} ∈ Vk
2
: |i−i0|+|j−j0|= 1
is called thek×k grid. In drawings ofGk, unless stated otherwise, for all i, j∈[k], the vertex (i, j) is drawn at the point (i, j) of a coordinate system, see Figure2.1. InGk, the vertex setCi ={(i, j): j∈[k]} is called the ith column for i ∈[k] and the vertex setRj = {(i, j): i ∈[k]} is called the jth row. An
(1,1) (5,1)
(1,5) (5,5)
horizontal edge in rowR2
vertical edge in columnC2
R1
C3
j
i
Figure 2.1: A 5×5 grid and its usual embedding.
edge{x, y} ∈Ek is called avertical edge in column Ci ifx∈Ciandy∈Ci. An edge{x, y} ∈Ek is called ahorizontal edge in rowRj ifx∈Rj andy∈Rj. Observe that each edge inEk is either horizontal or vertical.
When the names of the vertices inGkare not relevant, we often use the expression “ak×kgrid” in order to refer to a graph isomorphic to thek×kgridGk as defined above. We use the expression “k×kgrid”
to refer to the graph with the vertices and edges as defined above. A graphGisomorphic to thek×k grid for somek∈Nis also called asquare grid. Sometimes, non-square grids are used. More precisely, for two integersk, k0 ∈Nthek×k0 grid is defined to be the graphGk,k0 = (Vk,k0, Ek,k0) with
Vk,k0 :={(i, j): i∈[k], j∈[k0]} and Ek,k0 :=
{(i, j),(i0, j0)} ∈ Vk,k0
2
: |i−i0|+|j−j0|= 1 .
Roughly speaking, square grids are well-connected, meaning that many edges need to be removed in order to cut off a linear fraction of the vertices. This is made precise by the following results.
Lemma 2.5 (Theorem 6 in [LT79]).
For every k∈Nand every 0< β < 12, the following holds. Denote byGk = (Vk, Ek)the k×k grid and definen:=|Vk|=k2. If(B, W)is a cut inGk withβn≤ |B| ≤12n, theneGk(B, W)≥k·min1
2,√ β . Whereas the proof of the previous lemma is short and by a simple combinatorial argument, the following, stronger result has a long and involved proof.
Lemma 2.6 (edge isoperimetric inequalities, see [BL91]).
For every k ∈ N the following holds. If (B, W) is a cut in the k×k grid Gk with 14k2 ≤ |B| ≤ 34k2, theneGk(B, W)≥k.
Fix an integerk≥2. Withβ = 14 Lemma2.5implies that every bisection (B, W) in thek×kgrid cuts at least 12kedges. Lemma2.6is stronger and implies that every bisection (B, W) in thek×kgrid cuts at leastk edges. Another simple proof for a lower bound on the minimum bisection width of the square grid is also found in Chapter 1.9.1 in [Lei92].
Corollary 2.7.
For every integerk≥2 the k×k grid Gk satisfiesMinBis(Gk)≥k.
f1
f2
f3
f4
a)Example of the faces of a plane graph.
f1
f2
f3
f4
b)The vertices and edges on the boundary of the facef3 are colored blue.
Figure 2.2:A plane graph and its faces.
Planar Graphs
Roughly speaking, a drawing of a graphGis calledplanar if no two edges ofGcross and if two edges touch, then they touch in a common vertex. A graphGis calledplanar ifGadmits a planar drawing.
For a more formal definition of a planar graph as well as the following definitions, the reader is referred to Chapter 4 in [Die12]. A drawing of a graph is also called anembedding in the plane, or short anembedding as here no other surfaces are considered. Whenever an embedding or a drawing of a graph is considered here, we assume that it is planar. Aplanegraph is a graphGtogether with a drawing ofG. Consider a plane graphG. Then, the drawing ofGdivides the plane into several regions, which are calledfacesofG, see Figure2.2a). The following results for planar graphs are well-known.
Theorem 2.8 (Euler’s Formula, see Theorem 4.2.9 in [Die12]).
Every connected plane graph withn vertices,m edges, andf faces satisfies n−m+f = 2.
Corollary 2.9 (Corollary 4.2.10 in [Die12]).
Every planar graphG= (V, E)with|V| ≥3 satisfies|E| ≤3|V| −6.
Theorem 2.10 (Kuratowski’s Theorem, see Theorem 4.4.6 in [Die12]).
A graphGis planar if and only if Gdoes not contain a subgraph that is isomorphic to a subdivision ofK5
or a subdivision of K3,3.
Consider a plane graphG= (V, E), i. e., some embedding ofGis given. The boundary of a facef ofG is the set of pointspin the drawing ofGsuch that every open ball aroundpcontains a point inf and a point6=xthat is not inf. Here, points that represent a vertex ofGor belong to a polygon representing an edge ofGdo not belong to any face ofG. For a vertexv∈V that is embedded in the boundary of a facef, we say thatv is on the boundary off and for an edgee∈E that is embedded in the boundary of a facef, we say thateis on the boundary off. Figure2.2b)gives an example for the boundary of a face.