Intersection Graphs of Boxes and Cubes

Intersection graphs of boxes and cubes

A Thesis Submitted For the Degree of

Doctor of Philosophy in the Faculty of Engineering

Mathew C. Francis

Department of Computer Science and Automation Indian Institute of Science Bangalore – 560 012

July, 2009 To my parents and all my teachers Acknowledgements

Of all people, I should thank Dr. L. Sunil Chandran first, as the work behind this thesis is as much his as it is mine. The faith he reposed in me was at times as puzzling to me as it was reassuring. I am indebted to Dr. Naveen Sivadasan for the long discussions we had that not only produced results but went a long way in helping me learn the ropes. The brief but fruitful collaboration with Santhosh Suresh was thoroughly enjoyable. I am thankful to Dr. Samir Datta for his insights on planar graphs. The stimulating discussions with Dr. Irith Hartman, Rogers, Manu, Abhijin, Anita, Meghna, Sadagopan, Chintan and Subramanya have helped shape my view of the subject. Words cannot express my gratitude towards all my friends at IISc, each one of them inimitable, each one with a different perspective of the world but at the same time car- ing, guiding and helping with all their hearts. Rogers, Raj Mohan, Murali Sir, Sheron, Thomas, Ashik, Dileep, Shijo, Hari, Deepak Ravi, Rashid have all left indelible impres- sions on me. I am grateful to Nicky for her care and understanding. It is impossible to thank my parents enough for their unflinching support and constant encouragement.

i Abstract

A graph G is said to be an intersection graph of sets from a family of sets if there exists F a function f : V (G) such that for u, v V (G), (u, v) E(G) f(u) f(v) = → F ∈ ∈ ⇔ ∩ 6 . Interval graphs are thus the intersection graphs of closed intervals on the real line ∅ and unit interval graphs are the intersection graphs of unit length intervals on the real line. An interval on the real line can be generalized to a “k-box” in Rk.A k-box

B = (R1,R2,...,Rk), where each Ri is a closed interval on the real line, is defined to be the Cartesian product R R R . If each R is a unit length interval, we 1 × 2 ×···× k i call B a k-cube. Thus, 1-boxes are just closed intervals on the real line whereas 2-boxes are axis-parallel rectangles in the plane. We study the intersection graphs of k-boxes and k-cubes. The parameter boxicity of a graph G, denoted as box(G), is the minimum integer k such that G is an intersection graph of k-boxes. Similarly, the cubicity of G, denoted as cub(G), is the minimum integer k such that G is an intersection graph of k-cubes. Thus, interval graphs are the graphs with boxicity at most 1 and unit interval graphs are the graphs with cubicity at most 1. These parameters were introduced by F. S. Roberts in 1969. In some sense, the boxicity of a graph is a measure of how different a graph is from an interval graph and in a similar way, the cubicity is a measure of how different the graph is from a unit interval graph. We prove several upper bounds on the boxicity and cubicity of general as well as special classes of graphs in terms of various graph parameters such as the maximum degree, the number of vertices and the bandwidth. The following are some of the main results presented.

1. We show that for any graph G with maximum degree ∆, box(G) 2∆2. This ≤ ii iii

result implies that bounded degree graphs have bounded boxicity no matter how large the graph might be.

2. It was shown in [18] that the boxicity of a graph on n vertices with maximum degree ∆ is O(∆ln n). But a similar bound does not hold for the average degree

dav of a graph. [18] gives graphs in which the boxicity is exponentially larger than

dav ln n. We show that even though an O(dav ln n) upper bound for boxicity does

not hold for all graphs, for almost all graphs, boxicity is O(dav ln n).

3. The ratio of the cubicity to boxicity of any graph shown in [15] when combined with the results on boxicity show that cub(G) is O(∆ln2 n) and O(∆2 ln n) for any graph G on n vertices and with maximum degree ∆. By using a randomized construction, we prove the better upper bound cub(G) 4(∆ + 1) ln n . ≤ ⌈ ⌉ 4. Two results relating the cubicity of a graph to its bandwidth b are presented. First, it is shown that cub(G) 12(∆ + 1) ln(2b) +1. Next, we derive the upper bound ≤ ⌈ ⌉ cub(G) b + 1. This bound is used to derive new upper bounds on the cubicity of ≤ special graph classes like circular arc graphs, cocomparability graphs and AT-free graphs in relation to the maximum degree.

5. The upper bound for cubicity in terms of the bandwidth gives an upper bound of ∆ + 1 for the cubicity of interval graphs. This bound is improved to show that for any interval graph G with maximum degree ∆, cub(G) log ∆ + 4. ≤ ⌈ 2 ⌉ 6. Scheinerman [54] proved that the boxicity of any outerplanar graph is at most 2. We present an independent proof for the same theorem.

7. Halin graphs are planar graphs formed by adding a cycle connecting the leaves of a tree none of whose vertices have degree 2. We prove that the boxicity of any Halin graph is equal to 2 unless it is a complete graph on 4 vertices, in which case its boxicity is 1. Publications based on this thesis

1. “Geometric representation of graphs in low dimension using axis-parallel boxes”, L. Sunil Chandran, Mathew C. Francis and Naveen Sivadasan, accepted for publi- cation in Algorithmica, doi:10.1007/s00453-008-9163-5, 2008.

2. “Boxicity and maximum degree”, L. Sunil Chandran, Mathew C. Francis and Naveen Sivadasan, Journal of Combinatorial Theory, Series B, 98(2):443–445, March 2008.

3. “Representing graphs as the intersection of axis-parallel cubes”, L. Sunil Chandran, Mathew C. Francis and Naveen Sivadasan, MCDES 2008, Bangalore, May 2008.

4. “On the cubicity of AT-free graphs and circular-arc graphs”, L. Sunil Chandran, Mathew C. Francis and Naveen Sivadasan, Graph Theory, Computational Intelli- gence and Thought, Israel, September 2008.

5. “On the cubicity of interval graphs”, Graphs and Combinatorics, 25(2):169–179, May 2009.

6. “Boxicity of Halin graphs”, Discrete Mathematics, 309(10):3233–3237, May 2009.

iv Contents

Acknowledgements i

Abstract ii

Publications based on this thesis iv

1 Introduction 1 1.1 Basicdefinitionsandnotations...... 1 1.2 Intervalgraphsandboxicity ...... 3 1.2.1 k-boxes: intervals in higher dimensions ...... 5 1.2.2 Boxicity ...... 7 1.2.3 Interval graph representation of a graph ...... 8 1.3 Unitintervalgraphsandcubicity ...... 10 1.3.1 Unit and equal interval representations as mappings to real numbers 11 1.3.2 k-cubes ...... 12 1.3.3 Cubicity...... 13 1.3.4 Indifference graph representation of a graph ...... 13 1.4 Anoteontheasymptoticnotation ...... 14 1.5 Ashortsurveyofpreviousliterature ...... 14 1.5.1 Resultsonboxicity ...... 15 1.5.2 Boxicity in other scientific disciplines ...... 16 1.5.3 Resultsoncubicity ...... 17 1.5.4 Other geometric intersection graph classes ...... 18 1.6 Outlineoftherestofthethesis ...... 18

2 Upper bounds for boxicity 21 2.1 Previousupperboundsonboxicity ...... 21 2.1.1 Boxicity is O(∆ln n)...... 21 2.1.2 Boxicity and average degree ...... 22 2.2 Boxicity of bounded degree graphs ...... 22 2.3 Concludingremarks...... 24

v CONTENTS vi

3 Boxicity of random graphs 27 3.1 Randomgraphpreliminaries ...... 27 3.2 Boxicity is O(dav ln n)foralmostallgraphs...... 28 3.3 Remarks...... 31

4 A randomized construction for cubicity 33 4.1 The algorithm RAND ...... 34 4.2 Derandomizing RAND ...... 39 4.3 Ausefulresult...... 47

5 Cubicity and bandwidth 49 5.1 Cube representation in O(∆ln b)dimensions ...... 50 5.2 Cube representation in b +1dimensions ...... 55 5.3 Cubicity of special graph classes ...... 59 5.3.1 Circular-arcgraphs ...... 59 5.3.2 Cocomparabilitygraphs ...... 61 5.3.3 AT-freegraphs ...... 62 5.4 Asummaryofresults...... 63

6 Cubicity of interval graphs 65 6.1 Afewresultsthatweneed...... 65 6.2 Theproof ...... 67 6.3 Remarks...... 74

7 Planar graphs 77 7.1 Preliminaries ...... 77 7.2 Outerplanargraphs...... 79 7.3 Discussion...... 80

8 Boxicity of Halin graphs 81 8.1 Ashortintroduction ...... 81 8.2 Theproof ...... 82 8.3 Results...... 90

9 Conclusion 91 9.1 Improvements...... 91 9.2 Openproblems ...... 92 9.3 Endnote ...... 93

Bibliography 96 List of Figures

1.1 Anexampleofanintervalgraph...... 3 1.2 Anasteroidaltriple...... 4 1.3 A 2-box in R2 and a 3-box in R3 ...... 6 1.4 A 2-box representation for C4 ...... 7 1.5 K1,n, the star graph with n arms ...... 10

2.1 Structure of Gi ...... 23 5.1 Acircular-arcgraph ...... 59 5.2 Anexampleofacaterpillar ...... 62

7.1 A book drawing of K5 using3pages ...... 78 8.1 AHalingraph...... 81

vii Chapter 1

Introduction

All graphs considered in this work will be simple, undirected and finite. Most of the graph theoretic notations used shall be defined in the following section. Much of it has been borrowed from the book “Graph Theory” by Reinhard Diestel [26]. The reader may please refer to Chapter 1 of [26] for any notations that are not defined here.

1.1 Basic deﬁnitions and notations

The notations G(V,E), G =(V,E) or simply G will be used to indicate a graph G which has a vertex set V (G) and an edge set E(G). An edge between a vertex u and a vertex v will be denoted by (u, v) (or (v, u)) even though the edge is undirected. Thus, we will always assume that if (u, v) E(G), then (v, u) E(G). If(u, v) E(G), then u and ∈ ∈ ∈ v are adjacent in G; otherwise they are nonadjacent. A pair of vertices (u, v) E(G) is 6∈ said to be a non-edge or a missing edge in G. NG(u) is the neighbourhood of a vertex u in G, i.e., N (u) = v (u, v) E(G) . The degree of a vertex u in G, denoted G { | ∈ } by dG(u) is the number of vertices in G that are adjacent to u; or in other words, d (u) = N (u) . When there is no ambiguity about the graph under consideration, G | G | NG(u) and dG(u) might be abbreviated to N(u) and d(u) respectively. ∆(G) (or just ∆ if G is understood) will stand for the maximum degree of a vertex in G. The complement of a graph G, denoted by G is the graph with vertex set V (G) = V (G) and edge set

1 Chapter 1. Introduction 2

E(G) = (u, v) u, v V (G) and (u, v) E(G) . A graph H with V (H) V (G) and { | ∈ 6∈ } ⊆ E(H) E(G) is said to be a subgraph of G. A graph H is said to be an induced subgraph ⊆ of G if V (H) V (G) and E(H) = (u, v) E(G) u, v V (H) . One might also say ⊆ { ∈ | ∈ } that “H is the subgraph induced by V (H) in G” to indicate the same fact. A graph G′ is a supergraph of G if V (G)= V (G′) and E(G) E(G′). ⊆

Deﬁnition 1.1. If G1 and G2 are two graphs on the same vertex set V , we denote by G = G G the graph with vertex set V (G)= V and edge set E(G)= E(G ) E(G ). 1 ∩ 2 1 ∩ 2

G contains only those edges that are present in both G1 and G2. In other words, G1 and G2 are both supergraphs of G and every non-edge in G is a non-edge in either G1 or G2 or both. A path on n vertices, denoted by P , is the graph with vertex set V (P )= v ,v ,..., n n { 1 2 v and edge set E(P ) = (v ,v ) 1 i n 1 . A cycle on n vertices, denoted n} n { i i+1 | ≤ ≤ − } by C , is the graph with vertex set V (C ) = v ,v ,...,v and edge set E(C ) = n n { 1 2 n} n (v ,v ) 1 i n 1 (v ,v ) . { i i+1 | ≤ ≤ − } ∪ { n 1 } Given a graph G(V,E), a set of vertices S V (G) is said to be an independent set ⊆ if no two vertices in S are adjacent in G. On the other hand, a set of vertices S V (G) ⊆ is said to be a clique if every pair of vertices in S is adjacent in G. A graph G(V,E) is a complete p-partite graph if V (G) = A A A such 1 ∪ 2 ∪···∪ p that A is an independent set for each i and E(G)= (u, v) u A , v A and i = j . i { | ∈ i ∈ j 6 } If we let n = A , then we denote such a graph by K . We call each set A a i | i| n1,n2,...,np i “part”.

Deﬁnition 1.2. A permutation π on a ﬁnite set S is a bijection π : S 1, 2,..., S . → { | |}

Another way to think of π is as an ordering of the elements of the set S. A closed interval on the real line, denoted as [i, j] where i, j R and i j, is the ∈ ≤ set x R i x j . Given an interval X = [i, j], deﬁne l(X) = i and r(X) = j. { ∈ | ≤ ≤ } We say that the interval X has left end-point l(X) and right end-point r(X). Since we deal with only closed intervals throughout, we shall often shorten “closed interval” to Chapter 1. Introduction 3

just “interval”.

Deﬁnition 1.3. Let be a collection of sets. A graph G(V,E) is said to be an S intersection graph of sets from , if there is a function f : V (G) such that for S → S any two vertices u, v V (G), (u, v) E(G) f(u) f(v) = . ∈ ∈ ⇔ ∩ 6 ∅ In other words, it is possible to assign sets from to each vertex in G such that if S two vertices are adjacent, then the sets assigned to them have a non-empty intersection and if they are nonadjacent, the sets assigned to them are disjoint. Depending on what the collection is, one can deﬁne a variety of intersection graph S classes. For example, if is the collection of all closed intervals on the real line, the X class of intersection graphs of sets from is exactly the class of interval graphs. X

1.2 Interval graphs and boxicity

Deﬁnition 1.4. A graph G is an interval graph if f : V (G) u, v ∃ → X | ∀ ∈ V (G), (u, v) E(G) f(u) f(v) = , where is the set of all closed intervals ∈ ⇔ ∩ 6 ∅ X on the real line. The mapping f is called an interval representation of the graph G.

The examples below illustrate this concept.

[2, 3]

[1, 2] [1, 2]

[0, 1]

Figure 1.1: An example of an interval graph

An example of a graph which is not an interval graph is a chordless cycle on n vertices where n 4, denoted as C . The reason is easily explained as follows. Assume for the ≥ n Chapter 1. Introduction 4

sake of contradiction that Cn is indeed an interval graph. Then, there should exist an interval representation, say f, for Cn. Let x be the vertex in Cn whose interval has the leftmost left end-point. Let the cycle be xv1v2 ...vn−1x. Since (x, v2) is not an edge, the intervals f(x) and f(v2) are disjoint and since f(x) is the interval with the leftmost left end-point, we have r(f(x)) < l(f(v2)). For the same reason, we also have r(f(x)) < l(f(vn−2)) (note that v2 and vn−2 could be the same vertex if n = 4). It is easy to see that the interval of any vertex that is adjacent to both x and v2 or to both x and vn−2 will contain the point r(f(x)). Thus both the intervals f(v1) and f(vn−1) contain the point r(f(x)) implying that f(v ) f(v ) = . But (v ,v ) is not an 1 ∩ n−1 6 ∅ 1 n−1 edge in Cn thus contradicting our assumption that Cn is an interval graph. A cycle C in a graph G is an induced cycle if the subgraph induced by the vertices of C in G is C. In other words, the induced cycles in a graph are exactly the chordless cycles in that graph. Since any induced subgraph of an interval graph is also an interval graph, interval graphs cannot contain induced cycles of length more than 3.

Deﬁnition 1.5. A graph G is a chordal graph if there are no induced cycles of length more than 3 in it.

Interval graphs are thus a subclass of chordal graphs. But not all chordal graphs are interval graphs. Shown in Figure 1.2 is a graph that has no cycles (and hence is chordal) but is still not an interval graph.

v5 v3

v4 v6

Figure 1.2: v2, v4 and v6 form an asteroidal triple

An asteroidal triple (or AT in short) in a graph is an independent set of three vertices Chapter 1. Introduction 5

such that between any two of these vertices, there is a path in the graph that does not pass through any neighbour of the third vertex. It can be shown that an interval graph cannot contain an AT. Suppose G is an interval graph and the vertices x, y and z form an asteroidal triple in G. Let f be an interval representation of G. The intervals f(x), f(y) and f(z) are pairwise disjoint since x, y, z is an independent set. Assume without loss { } of generality that the interval f(y) is in between f(x) and f(z). Now, it is not diﬃcult to convince oneself that any path in G between x and z will contain at least one vertex v such that f(v) overlaps f(y). This contradicts the fact that x, y, z is an asteroidal { } triple in G.

The graph in Figure 1.2 is not an interval graph because the vertices v2, v4 and v6 form an asteroidal triple.

Deﬁnition 1.6. A graph G is an AT-free graph if it contains no asteroidal triples.

It turns out that the two concepts of large induced cycles and asteroidal triples are enough to characterize interval graphs. If a graph does not have induced cycles of length more than 3 or asteroidal triples in it, then it is an interval graph.

Theorem 1.7 (Lekkerkerker and Boland [43]). A graph is an interval graph if and only if it is chordal and AT-free. The reader should note that Deﬁnition 1.4 can be changed to use open intervals instead of closed intervals. It is an easy exercise to prove that the class of intersection graphs of open intervals on the real line is the same as that of closed intervals and therefore, a separate treatment of the two is unnecessary.

1.2.1 k-boxes: intervals in higher dimensions

An interval is the collection of all points on the real line between an upper and a lower bound. How can we generalize this notion to higher dimensional spaces, say to R2, from the real line? We could look at an ordered pair of intervals of the form (Ix,Iy). Note that 2 an ordered pair of intervals (Ix,Iy) describes a rectangle in R (with its sides parallel to the axes) as shown in Figure 1.3. In other words, (I ,I ) denotes the set I I of x y x × y Chapter 1. Introduction 6

2 points in R . It is easy to see that given two rectangles A =(A1,A2) and B =(B1, B2),

Y Y

Iy B =(Ix,Iy) B =(Ix,Iy,Iz)

Iy Z Iz X Ix X Ix

Figure 1.3: A 2-box in R2 and a 3-box in R3

A B = (i.e., the two rectangles have at least one point in common) if and only if if ∩ 6 ∅ there is an overlap between intervals A1 and B1 (on the X-axis) and between intervals

A2 and B2 (on the Y -axis). We call these rectangles 2-boxes, in the sense that they are boxes in the 2-dimensional plane R2. We can generalize this deﬁnition to k dimensions by deﬁning the notion of a k-box.

Deﬁnition 1.8. A k-box, denoted as a k-tuple of intervals (R1,R2,...,Rk) is the set of points R R R . 1 × 2 ×···× k A k-box could be thought of as a “k-dimensional box” or a “box” in Rk with its sides parallel to the axes. We sometimes refer to such boxes as “axis-parallel k-dimensional boxes”. Given two k-boxes A = (A ,...,A ) and B = (B ,...,B ), A B = 1 k 1 k ∩ 6 ∅ ⇔ i 1 i k, A B = . ∀ | ≤ ≤ i ∩ i 6 ∅ Since a k-box denoted by a k-tuple of intervals, k denotes the set of all k-boxes. X A graph G is said to be an intersection graph of k-boxes if there exists a mapping f : V (G) k such that (u, v) E(G) f(u) f(v) = . Such a mapping f is called →X ∈ ⇔ ∩ 6 ∅ a k-box representation of G. Let us denote by , the class of intersection graphs of k- Hk boxes, or in other words, the class of graphs that have k-box representations. If a graph G , we say that G is “representable” or “can be represented” as the intersection of ∈Hk Chapter 1. Introduction 7

k-boxes. By our deﬁnition of a k-box, a 1-box is just an interval on the real line. Thus, is exactly the class of interval graphs. Further, it can be easily seen that for j>i, H1 . This is because if a graph G has an i-box representation f, then it also has Hi ⊆ Hj a j-box representation g which can be deﬁned as follows: for every vertex u V (G), ∈ g(u) is obtained by appending an arbitrary interval I, (j i) times to the i-tuple f(u). − Thus, if f(u)=(f1(u),...,fi(u)), then g(u)=(f1(u),...,fi(u),I1,I2,...,Ij−i) where I = I = = I = I and I is an arbitrary interval. 1 2 ··· j−i But does using higher dimensional boxes give us more power? Do more graphs become representable as the intersection of k-boxes as we increase k? Let us consider the class

of intersection graphs of 2-boxes. The graph C4, that was observed to be not an interval graph can be seen to be an intersection graph of 2-boxes (see Figure 1.4). This example

v1 v1 v4 v v v2 4 2 v3

Figure 1.4: A 2-box representation for C4

shows that . H1 ⊂H2

1.2.2 Boxicity

We are now ready to deﬁne the parameter boxicity of a graph.

Deﬁnition 1.9. The boxicity of a graph G, denoted as box(G), is the minimum positive integer k such that G is representable as the intersection of k-boxes.

Thus, G is an interval graph if and only if box(G) = 1. Also, since C4 is not an

interval graph but has a 2-box representation as we have seen above, box(C4) = 2. Chapter 1. Introduction 8

The natural question to ask now is how high can the boxicity of a graph be? Will it even be ﬁnite? It can be easily shown that if G is any graph on n vertices, box(G) n. ≤ In fact, a slightly more careful analysis shows that box(G) n/2 for any graph G on ≤ ⌊ ⌋ n vertices. Roberts [51] has shown that a complete n/2-partite graph with 2 vertices in each part has boxicity equal to n/2. This graph, which we call the Roberts’ graph on n vertices is just a complete graph on n vertices with a maximum matching removed from it. This also shows that for any k N, and k 1, there exists a graph with boxicity ∈ ≥ equal to k, namely the Roberts’ graph on 2k vertices. It can thus be concluded that for any k, since the Roberts’ graph on 2(k + 1) vertices is in but not in Hk ⊂ Hk+1 Hk+1 . Hk

1.2.3 Interval graph representation of a graph

Below, we state a very useful lemma due to Roberts [51].

Lemma 1.10 (Roberts [51]). For any graph G, box(G) k if and only if there exists ≤ k interval graphs I ,...,I such that G = I I . 1 k 1 ∩···∩ k Proof: ( ): If box(G) k then there exists a function f : V (G) k such that for ⇒ ≤ → X any u, v V (G), (u, v) E(G) f(u) f(v) = . Deﬁne functions f ,...,f on ∈ ∈ ⇔ ∩ 6 ∅ 1 k V (G) as follows: for u V (G), f(u)=(f (u),...,f (u)). For 1 i k, let I be an ∈ 1 k ≤ ≤ i interval graph with vertex V (G) and interval representation f . Now, (u, v) E(G) i ∈ ⇔ f(u) f(v) = i, f (u) f (v) = i, (u, v) E(I ). It now follows that ∩ 6 ∅ ⇔ ∀ i ∩ i 6 ∅ ⇔ ∀ ∈ i G = I I . 1 ∩···∩ k ( ): Let G = I I . For 1 i k, let f : V (G) be an interval ⇐ 1 ∩···∩ k ≤ ≤ i → X representation for I (recall that V (I ) = V (G)). Deﬁne f : V (G) k as follows: i i → X for u V (G), f(u)=(f (u),...,f (u)). We claim that f is a k-box representation for ∈ 1 k G. Since G = I I , (u, v) E(G) i, (u, v) E(I ) i, f (u) f (v) = 1 ∩···∩ k ∈ ⇔ ∀ ∈ i ⇔ ∀ i ∩ i 6 f(u) f(v) = . f is therefore a k-box representation for G thus proving that ∅ ⇔ ∩ 6 ∅ box(G) k. ≤ Chapter 1. Introduction 9

Note that the interval graphs I1,...,Ik are supergraphs of G. Thus, the forward implication of the lemma means that if box(G) k, then it is possible to ﬁnd k interval ≤ supergraphs of G such that every edge that is not present in G is not present in at least one of these interval supergraphs. Conversely, if one can ﬁnd k interval graphs I1,...,Ik such that G = I I , then box(G) k. Given below is a straightforward corollary 1 ∩···∩ k ≤ of Lemma 1.10.

Corollary 1.11. If G = G G G , then box(G) k box(G ). 1 ∩ 2 ∩···∩ k ≤ i=1 i Deﬁnition 1.12. A collection of interval graphs such that theirP intersection gives the graph G is said to be an interval graph representation of G.

Almost always, we prove that the boxicity of a given graph G is not more than k by constructing an interval graph representation of G with k interval graphs. As an example, we prove a claim that we made earlier.

Theorem 1.13 (Roberts [51]). If G is any graph on n vertices, box(G) n. ≤ Proof: For u V (G), let I be an interval graph with vertex set V (G) and interval ∈ u representation fu given by:

fu(u) = [0, 1], v N(u), f (v) = [1, 2], and ∀ ∈ u v N(u), f (v) = [2, 3]. ∀ 6∈ u It can be easily veriﬁed that I u V (G) is an interval graph representation of { u | ∈ } G with n interval graphs. It now follows from Lemma 1.10 that box(G) n. ≤

It should be noted that if H is an induced subgraph of G, then box(H) box(G). ≤ This is because if fG is a k-box representation for G, then one can obtain a k-box representation f for H by letting f = f , the restriction of f to V (H). This H H G|V (H) G observation also means that the boxicity of any graph is greater than or equal to the boxicity of any of its induced subgraphs. When we deal with box representations of graphs, we are free to use boxes of arbitrary dimensions, that is to say that the boxes assigned to two diﬀerent vertices need not be Chapter 1. Introduction 10

of the same size or shape as long as they are both axis-parallel. It seems worthwhile to think about more restricted box representations. For example, what if want all the boxes used in a box representation to have the same size (i.e., the same dimensions)? Can such a representation in box(G) dimensions be obtained for every graph G? Let us look at the simplest case first—when box(G) = 1. The question posed above is equivalent to asking whether for an interval graph G, there exists an interval representation such that the intervals assigned to each vertex are of the same length (we define the “length” of an interval [x , x ] to be x x ). The answer is no, as illustrated by the graph K , also 1 2 2 − 1 1,n known as the star graph (shown in Figure 1.5). K1,n is an interval graph as it has an interval representation as shown in the figure. But some observation can convince the reader that if n 3, K cannot have an interval representation in which all the vertices ≥ 1,n are assigned intervals of the same length. Some interval graphs (like the one shown in

v3 v2 v1 v2 v3 vn−2 vn−1 vn . . . v 1 c c vn v n−2 vn−1

Figure 1.5: K1,n, the star graph with n arms and an interval representation for it

Figure 1.1) do have interval representations that assign intervals of the same length to each vertex. As we see in the next section, the class of such interval graphs are called unit interval graphs, proper interval graphs or indiﬀerence graphs.

1.3 Unit interval graphs and cubicity

Now, if we let to be the set of all unit length intervals on the real line, the class X1 ⊂X of intersection graphs on is the class of unit interval graphs or indiﬀerence graphs. X1 Chapter 1. Introduction 11

Deﬁnition 1.14. A graph G is a unit interval graph if f : V (G) u, v ∃ → X1 | ∀ ∈ V (G), (u, v) E(G) f(u) f(v) = , where is the set of all closed intervals of ∈ ⇔ ∩ 6 ∅ X1 length 1 on the real line. The mapping f is called a unit interval representation of the graph G.

For x R+, let denote the set of all closed intervals of length x on the real line. If ∈ Xx a graph G has a unit interval representation f, then for any x R+ it also has an interval ∈ representation g : V (G) deﬁned as: u V (G), g(u) = [x l(f(u)), x r(f(u))]. → Xx ∀ ∈ · · Clearly, g is an interval representation for G that maps the vertices in G to intervals of length x. g is thus an equal interval representation as deﬁned below.

Deﬁnition 1.15. An interval representation f of a graph G is called an equal interval representation with interval length x if for each v V (G), r(f(v)) l(f(v)) = x. ∈ − Conversely, if a graph G has an equal interval representation g with interval length x (where x R+), then it has a unit interval representation f given by: u V (G), f(u)= ∈ ∀ ∈ 1 l(g(u)), 1 r(g(u)) . It can thus be seen that unit interval graphs are exactly those x · x · graphs with equal interval representations. Note that the class of unit interval graphs is also exactly the class of interval graphs which have an interval representation such that the interval assigned to no vertex is properly contained in the interval assigned to another vertex as shown in [32]. Therefore, these graphs are also called proper interval graphs.

1.3.1 Unit and equal interval representations as mappings to real numbers

Since a unit length interval is completely specﬁed by just one of its end-points, a unit interval representation could assign just real numbers (instead of unit length intervals) to vertices in such a way that two vertices are adjacent if and only if the real numbers assigned to them diﬀer by at most 1. Note that we could think of these real numbers as the left end-points of the unit intervals assigned to the vertices. The same is true for equal interval representations of interval length x. In this case, two vertices are Chapter 1. Introduction 12

adjacent if and only if the real numbers assigned to them differ by at most x. This idea can be expressed mathematically as follows. If f is an equal interval representation with interval length x for the unit interval graph G, then define g : V (G) R as: for → u V (G), g(u)= l(f(u)). Now, (u, v) E(G) f(u) f(v) = g(u) g(v) x. ∈ ∈ ⇔ ∩ 6 ∅ ⇔ | − | ≤ Conversely, if g is a function that maps the vertices of G to real numbers such that (u, v) E(G) g(u) g(v) x for some x R+, then we can define a function ∈ ⇔ | − | ≤ ∈ f : V (G) as: for u V (G), f(u) = [g(u),g(u)+ x]. We therefore have (u, v) → Xx ∈ ∈ E(G) g(u) g(v) x f(u) f(v) = . We implicitly assume the existence of ⇔ | − | ≤ ⇔ ∩ 6 ∅ f when we speak of g and therefore we do not make any distinction between f and g. Thus, we say that g is an equal interval representation with interval length x for G and if x = 1, we say that g is a unit interval representation for G. We thus have the following alternate definition for unit and equal interval representations.

Deﬁnition 1.16. Given a graph G, a function f : V (G) R such that (u, v) E(G) → ∈ ⇔ f(u) f(v) x is called an equal interval representation with interval length x of | − | ≤ the graph G. If x = 1, then we call f a unit interval representation of G.

1.3.2 k-cubes

Recall that we generalized intervals on the real line to k-boxes in Rk. Along the same lines, we deﬁne a k-cube as follows.

Deﬁnition 1.17. A k-cube, denoted as (R1,R2,...,Rk), where each Ri is a unit length interval on the real line, is the set of points R R R . 1 × 2 ×···× k k-cubes are also referred to as “axis-parallel k-dimensional cubes”. Since a k-cube is denoted by a k-tuple of unit length intervals, it can be thought to be a member of the set ( )k. As we saw in the last paragraph, each R , being a unit interval, is X1 i completely deﬁned by just specifying its left end-point l(Ri), since r(Ri) = l(Ri) + 1.

Thus the k-cube (R1,R2,...,Rk) can be alternately denoted by a k-tuple of real numbers k (l(R1), l(R2),...,l(Rk)). This notation allows us to think of k-cubes as members of R Chapter 1. Introduction 13

and often makes their handling easier. If A, B Rk are two k-cubes such that A = ∈ (a ,...,a ) and B =(b , . . . , b ), then A B = if and only if for each i, a b 1. 1 k 1 k ∩ 6 ∅ | i − i| ≤ A graph G is said to be an intersection graph of k-cubes if f : V (G) Rk, such ∃ → that (u, v) E(G) f(u) f(v) = or in other words, there exists a mapping f that ∈ ⇔ ∩ 6 ∅ maps the vertices of G to k-cubes such that two vertices u and v in G are adjacent if and only if the k-cubes corresponding to them have a non-empty intersection. Such a mapping f is called a k-cube representation of G.

1.3.3 Cubicity

Deﬁnition 1.18. The cubicity of a graph G, denoted by cub(G), is deﬁned to be the minimum integer k such that G has a k-cube representation.

The graphs with cubicity 1 are therefore exactly the class of unit interval graphs. The cubicity of any graph on n vertices is at most 2n/3 as shown by Roberts in [51]. He also shows that the Roberts’ graph on n vertices has cubicity equal to 2n/3. Note that since a k-cube is also a k-box, any graph that is an intersection graph of k-cubes is also an intersection graph of k-boxes. From this observation, it follows that for any graph G, box(G) cub(G). ≤

1.3.4 Indiﬀerence graph representation of a graph

Recall that unit interval graphs are also called indiﬀerence graphs. Similar to Lemma 1.10 for boxicity, we have the following lemma for cubicity.

Lemma 1.19 (Roberts [51]). For any graph G, cub(G) k if and only if there exists ≤ k indifference graphs (unit interval graphs) I ,...,I such that G = I I . 1 k 1 ∩···∩ k A collection of indifference graphs whose intersection gives the graph G is called an indifference graph representation or a unit interval graph representation of G. Thus in order to prove that a graph G has cubicity at most k, we just need to produce an indifference graph representation of G using k indifference graphs. Akin to that for boxicity, we have the following corollary to Lemma 1.19. Chapter 1. Introduction 14

Corollary 1.20. If G = G G G , then cub(G) k cub(G ). 1 ∩ 2 ∩···∩ k ≤ i=1 i P 1.4 A note on the asymptotic notation

We use the asymptotic notation to express various bounds on the boxicity and cubicity of graphs. A short description of the way in which we use the asymptotic notation is given below. Let ℘(G) be a graph parameter such as box(G) or cub(G) and let f(G) be a function that is deﬁned in terms of various parameters of G. We denote by “℘(G) = O(f)” or “℘(G) O(f)” or “℘ is O(f)” the fact that there exists constants c and c such that ∈ 0 for any graph G, ℘(G) c + cf(G). For example, in Chapter 4, we prove that for ≤ 0 any graph G on n vertices and having maximum degree ∆, cub(G) 4(∆ + 1) ln n . ≤ ⌈ ⌉ Because of this result, we say that cub(G) is O(∆ln n). Similarly, by “℘(G) = Ω(f)” or “℘(G) Ω(f)” or “℘ is Ω(f)” we mean that there ∈ exist constants c and c such that f(G) c + c℘(G). “℘(G)=Θ(f)” denotes the fact 0 ≤ 0 that ℘(G)= O(f) and ℘(G)=Ω(f).

1.5 A short survey of previous literature

The parameters boxicity and cubicity of graphs were introduced by F. S. Roberts [51] in 1969. Roberts showed that for any graph G on n vertices box(G) n/2 and cub(G) ≤ ≤ 2n/3. Both these bounds are tight since box(K2,2,...,2) = n/2 and cub(K3,3,...,3) = 2n/3 where K2,2,...,2 denotes the complete n/2-partite graph with 2 vertices in each part and

K3,3,...,3 denotes the complete n/3-partite graph with 3 vertices in each part. It is easy to see that the boxicity of any graph is at least the boxicity of any induced subgraph of it. Chapter 1. Introduction 15

1.5.1 Results on boxicity

It was shown by Cozzens [23] that computing the boxicity of a graph is NP-hard. This was later improved by Yannakakis [62], and ﬁnally by Kratochv´ıl [40] who showed that deciding whether the boxicity of a graph is at most 2 itself is NP-complete. In many algorithmic problems related to graphs, the availability of certain convenient representations turns out to be extremely useful. Probably, the most well-known and important examples are the tree decompositions and path decompositions [7]. Many NP-hard problems are known to be polynomial time solvable given a tree(path) decom- position of bounded width for the input graph. Similarly, the representation of graphs as intersections of “disks” or “spheres” lies at the core of solving problems related to frequency assignments in radio networks, computing molecular conformations etc. For the maximum independent set problem which is hard to approximate within a factor of n(1/2)−ǫ for general graphs [35], a PTAS is known for disk graphs given the disk representation [27, 13]. In a similar way, the availability of a box representation in low dimension makes some well known NP hard problems polynomial time solvable. For example, it was shown in [53] that the max-clique problem is polynomial time solvable in graph classes with a polynomial bound on the number of maximal cliques. Since boxicity k graphs have only O((2n)k) maximal cliques, the max-clique problem admits a polynomial-time algorithm in bounded boxicity graphs. It was shown in [35] that the complexity of ﬁnding the maximum independent set is hard to approximate within a factor n(1/2)−ǫ for general graphs. In fact, [35] gives the stronger inapproximability result of n1−ǫ, for any ǫ > 0, under the assumption that NP=ZPP. Though this problem is 6 NP-hard even for boxicity 2 graphs, it is approximable to a factor of 1+ 1 log n d−1 for ⌊ c ⌋ any constant c 1 for boxicity d (d 2) graphs given a box representation [2, 5]. It was ≥ ≥ shown in [16] that for any graph G, box(G) tw(G) + 2, where tw(G) is the treewidth ≤ of G. This result implies that the class of ‘low boxicity’ graphs properly contains the class of ‘low treewidth graphs’. Researchers have also tried to bound the boxicity of graph classes with special structure. Scheinerman [54] showed that the boxicity of outer planar graphs is at most 2. Chapter 1. Introduction 16

Thomassen [57] proved that the boxicity of planar graphs is bounded above by 3. Upper bounds for the boxicity of many other graph classes such as chordal graphs, AT-free graphs, permutation graphs etc. were shown in [16] by relating the boxicity of a graph with its treewidth. Researchers have also tried to generalize or extend the concept of boxicity in various ways. The poset boxicity [59], the rectangle number [20], grid dimension [4], circular dimension [30, 55] and the boxicity of digraphs [19] are some examples.

1.5.2 Boxicity in other scientiﬁc disciplines

Box representations of graphs ﬁnd application in problems from ecology and operations research. As an example, we give an outline of a problem from ecology below:

Niche problem in ecology:

Ecologists study the interactions between various organisms in an environment. Each species has a natural habitat in which it is commonly found. If we examine different environmental factors like temperature, humidity, pH etc. of the natural habitats of a species, we can find for each factor a range of values which characterizes the habitats in which the species is found. If we have k such factors, we can define a k-dimensional space with an axis for each such factor. Such a space is called the “ecological phase space”. The range of values of each factor for a species together defines a k-box, or the “ecological niche” of the species. If the ecological niches of two species overlap, then they can be together found in some habitats. Ecologists traditionally use directed graphs called “food webs” which define the “predator-prey” relationship between a set of species. There is an edge from a species X to a species Y in this graph if Y preys on X. Now, two species compete for food if they have a common prey. An undirected graph drawn with the vertex set as a set of species and with edges in such a way that there is an edge between two species if they have a common prey is called a “competition graph”. An edge in this graph between two species X and Y means that X and Y compete for food. At the same time, considering an ecological phase space in which one dimension is the “feeding dimension” (an axis with the kinds of food that various species eat along Chapter 1. Introduction 17

it), two species compete if and only if their ecological niches in this phase space overlap. Now, if we have a competition graph of a set of species from various sources of data like food webs, then the question of what the boxicity of the graph is becomes interesting. This problem was studied extensively by Cohen [21]. He observed that in most cases, the competition graphs turn out to be interval graphs which means that one dimension suﬃces to explain the competition graph. Considering that a large majority of possible graphs are not interval graphs, this seems too much of a coincidence. Roberts gives a nice overview of this problem in [52].

1.5.3 Results on cubicity

It has been shown that deciding whether the cubicity of a given graph is at least 3 is NP-hard [62]. It is easy to see that the problem of representing graphs using k-cubes can be equiv- alently formulated as the following geometric embedding problem. Given an undirected unweighted graph G = (V,E) and a threshold t, ﬁnd an embedding f : V Rk of the → vertices of G into a k-dimensional space (for the minimum possible k) such that for any two vertices u and v of G, f(u) f(v) t if and only if u and v are adjacent. The || − ||∞ ≤ norm is the L norm. Clearly, a k-cube representation of G yields the required || ||∞ ∞ embedding of G in the k-dimensional space. The minimum dimension required to embed

G as above under the L2 norm is called the sphericity of G. Refer to [47] for applications where such an embedding under L∞ norm is argued to be more appropriate than embedding under L2 norm. The connection between cubicity and sphericity of graphs were studied in [31, 45]. The cube representation of special classes of graphs like hypercubes and complete multipartite graphs were investigated in [51, 45, 48]. Also, the cubicity of

d the d-dimensional hypercube was shown to be Θ( log d ) in [17]. A lower bound for the cubicity of general graphs in terms of the diameter and maximum independent set size was shown in [14]. The ratio of cubicity to boxicity of any graph on n vertices was shown to be at most log n in [15]. ⌈ 2 ⌉ Chapter 1. Introduction 18

1.5.4 Other geometric intersection graph classes

Like interval and unit interval graphs, a number of classes of geometric intersection graphs have been studied. Circular arc graphs [32] are the intersection graphs of arcs on a circle and circle graphs are the intersection graphs of chords of a circle. Tolerance graphs [33] generalize interval graphs to allow a restricted overlap between two intervals. An intersection model for permutation graphs is given [32]. Trapezoid graphs are the intersection graphs of trapezoids between two parallel lines [25]. Graphs deﬁned as the intersection of a number of diﬀerent kinds of geometric objects in the plane are described in [40]. Interval catch digraphs [49] have an intersection model very similar to that of interval graphs but are directed graphs. In this model, a pair (Ix,px), where Ix is an interval and px is a point in Ix, is assigned to each vertex such that there is a directed edge (x, y) in the graph if and only if p I . y ∈ x Another generalization of interval graphs is to make the set assigned to each vertex the union of k intervals such that two vertices are adjacent if and only if the sets assigned to them have a non-empty intersection. The minimum k required to represent a graph in such a way is called its interval number [58]. A survey of a number of intersection graph classes and their applications is available in [46].

1.6 Outline of the rest of the thesis

Chapter 2 investigates the relationship between the maximum degree and the boxicity of a graph. The previous upper bound for boxicity in terms of the maximum degree ∆ of a graph was (∆ + 2)ln n 1. A new upper bound of 2∆2 for boxicity is presented, ⌈ ⌉ thereby showing that the boxicity of a bounded degree graph is bounded no matter how large the graph is.

1Note that almost invariably, we use n to denote the number of vertices of the graph under consideration. Chapter 1. Introduction 19

Chapter 3 shows that even though there are graphs whose boxicity is not O(dav ln n) where dav is the average degree, such graphs are rare. The theory of random graphs is used to show that in a suitable random graph model, the probability of the randomly drawn graph to have a boxicity that is O(dav ln n) goes to 1 as n becomes large. We make use of the upper bound on boxicity proved in Chapter 2 to prove this result. In Chapter 4, we see that if we randomly generate 4(∆ + 1) ln n indifference su- ⌈ ⌉ pergraphs of an input graph G, then there is a slight possibility that these indifference graphs form an indifference graph representation of G. Thus we have an upper bound of 4(∆ + 1) ln n on the cubicity of a graph. The randomized algorithm is derandomized ⌈ ⌉ to obtain a deterministic polynomial-time algorithm that outputs a cube representation of the input graph in 4(∆ + 1) ln n dimensions. ⌈ ⌉ Two results relating the cubicity and the bandwidth of a graph are presented in Chapter 5. A bandwidth ordering of the graph is taken as input and the construction introduced in Chapter 4 is applied to show an O(∆ln b) upper bound for the cubicity of any graph with maximum degree ∆ and bandwidth b. Another upper bound of b +1 on the cubicity is also shown. This bound is used to show upper bounds on the cubicity of circular-arc graphs, cocomparability graphs and AT-free graphs. Each of Chapters 6–8 deals with a special graph class. The upper bound of b+1 for cubicity automatically gives us an upper bound of ∆+1 for the cubicity of any interval graph. In Chapter 6, we show that a much tighter upper bound of log ∆ + 4 exists for the cubicity of interval graphs. ⌈ 2 ⌉ Outerplanar graphs are studied next. As mentioned before, it was proved by Schein- erman [54] that outerplanar graphs need boxicity at most 2. Chapter 7 gives an independent proof that shows the same result. In Chapter 8, we look at Halin graphs, which are a restricted class of planar graphs incomparable with the class of outerplanar graphs. We show that every Halin graph that is not a K4 has boxicity equal to 2.

Chapter 2

Upper bounds for boxicity

Roberts gave us an upper bound of n/2 for the boxicity of any graph on n vertices. We shall now try to derive a diﬀerent upper bound for boxicity in terms of the maximum degree ∆ of the graph.

2.1 Previous upper bounds on boxicity

2.1.1 Boxicity is O(∆ ln n)

Lemma 1.10 tells us that the boxicity of a graph G is the minimum number of interval supergraphs of G such that each non-edge (or “missing edge”) in G is a non-edge in at least one of these interval supergraphs. One could try to devise some method by which we can obtain supergraphs of G in such a way that each missing edge in G is missing in one of these supergraphs. Of course, one could obtain supergraphs of G by adding arbitrary sets of edges to G. But the catch is that we need only those supergraphs of G that are also interval graphs. It seems diﬃcult to systematically generate supergraphs of G that are also interval graphs. In [18], Chandran and Sivadasan try to generate interval supergraphs of G at random and come up with a simple randomized algorithm that generates an interval graph representation of the input graph G on n vertices and with maximum degree ∆ using (∆+1)ln n interval graphs with non-zero probability. ⌈ ⌉ The existence of this algorithm proves the following theorem.

21 Chapter 2. Upper bounds for boxicity 22

Theorem 2.1 (Chandran and Sivadasan). Given a graph G on n vertices with maximum degree ∆, box(G) (∆+2)ln n . ≤ ⌈ ⌉ In Chapter 4, we extend this randomized construction to show that a similar upper bound exists for cubicity.

2.1.2 Boxicity and average degree

The relationship between the boxicity of a graph and its average degree is also explored in [18]. It is shown that in general the boxicity of a graph on n vertices with average degree dav is not O(dav ln n) as there exist graphs with boxicity that is exponentially larger than dav ln n. In Chapter 3, we show that even though such graphs exist, for most graphs, boxicity is O(dav ln n).

2.2 Boxicity of bounded degree graphs

If the family of graphs under consideration has bounded degree, the upper bound of (∆+2)ln n for the boxicity is an improvement over previous bounds as it implies that ⌈ ⌉ boxicity of graphs in that family is O(ln n). But is this the best possible for bounded degree graphs? No matter what graph we take, it seems that the boxicity is always less than or equal to ∆. Might it be the case that the boxicity of any graph with maximum degree ∆ is O(∆)? The anwer to that question certainly does not appear to be easy. We could ﬁrst try and see if boxicity can be bounded from above by a function of ∆ alone. Such an upper bound would be interesting as it would mean that the boxicity of graphs with bounded degree—like expander graphs—is bounded no matter how large the graph is. We shall now look at a simple proof that shows that boxicity is in fact O(∆2). In order to avoid confusion, we shall use ∆(G) to denote the maximum degree of a graph G for the remainder of this section. We shall show that for any graph G with maximum degree ∆(G), box(G) 2∆(G)2. Let χ(G) denote the chromatic number of ≤ G. We use Brooks’ theorem, which states that χ(G) ∆(G) for any connected graph G ≤ unless it is an odd cycle or a complete graph. We also use the square G2 of a graph G, Chapter 2. Upper bounds for boxicity 23

deﬁned to be the graph obtained from G by adding edges joining nonadjacent vertices that have a common neighbour in G. Note that since any vertex will become adjacent to at most ∆(G)(∆(G) 1) new vertices when the graph is squared, ∆(G2) ∆(G)2. − ≤ If G = G G G , then by Corollary 1.11, box(G G ) k box(G ); 1 ∩ 2 ∩···∩ k 1 ∩···∩ k ≤ i=1 i we will use this fact. P

Theorem 2.2. If G is a graph with ∆(G)= D, then box(G) 2D2. ≤ Proof: Let n = V (G) . Let k = χ(G2), and let c be a proper k-coloring of G2 using colors | | 1,...,k. For 1 i k, let V = u V (G) : c(u)= i (recall that V (G2)= V (G)). For ≤ ≤ i { ∈ } 1 i k, let H be the complete graph with vertex set V (G) V , and let G be the ≤ ≤ i − i i graph with V (G )= V (G) and E(G )= E(G) E(H ). i i ∪ i Consider vertices u and v. If they are adjacent in G, then they are adjacent in each G , since E(G) E(G ). If they are not adjacent in G, then they are nonadjacent in i ⊆ i both G and G . Hence G = G G . Note that G2 will contain a triangle c(u) c(v) 1 ∩···∩ k if there is a vertex with degree 2 or more in G. Therefore, it is clear that G2 cannot be an odd cycle except when n = 3, in which case it is a complete graph. If G2 is a complete graph, we have D2 ∆(G2)= n 1 and therefore, box(G) 2D2 (because we ≥ − ≤ know that box(G) n/2). Thus, by Brooks’ theorem, we can assume that k = χ(G2) ≤ ≤ ∆(G2) D2. Now, it suﬃces to show that box(G ) 2 for each i. ≤ i ≤

Vi · · ·

= Gi V Vi −

Figure 2.1: Structure of Gi: the two dotted edges cannot be both present Chapter 2. Upper bounds for boxicity 24

If x, y V (i.e., c(x)= c(y)= i) and (x, w), (y,w) E(G) for some w V (G), then ∈ i ∈ ∈ (x, y) E(G2), which prevents c(x) = c(y). Hence in G, each vertex outside V has at ∈ i most one neighbour in Vi (see Figure 2.1). By construction, the edges of Gi incident to

Vi are edges of G. Hence in Gi each vertex outside Vi has at most one neighbour in Vi. To obtain box(G ) 2, we deﬁne interval graphs I and I′ on V (G) whose intersection i ≤ is G . Let V = v ,...,v . In both I and I′, assign the single-point interval j to v . i i { 1 h} { } j Consider w V (G) V . If w has no neighbour in V , then assign w the single-point ∈ − i i intervals 0 in I and n in I′. If w has neighbour v V (there can only be one such { } { } j ∈ i neighbour, as noted before), then assign w the intervals [0, j] in I and [j, n] in I′. By construction, E(G ) E(I) E(I′). i ⊆ ∩ ′ It remains to show that nonadjacent vertices in Gi are nonadjacent in I or I . All nonadjacent pairs in G include a vertex of V ; consider v V . Let (v ,w) be the i i j ∈ i j ′ nonadjacent pair. Note that Vi is independent in both I and I . Thus, we can assume that w V (G) V . Then either the interval for w in I ends before the point j, or the ∈ − i interval for w in I′ begins after the point j.

2.3 Concluding remarks

We have seen that the availability of a low dimensional box representation for a graph can lead to polynomial time algorithms and to better approximation ratios for NP-hard problems. Thus, it is interesting to design eﬃcient algorithms to represent graphs of small boxicity in a small number of dimensions. Theorem 2.2 gives an upper bound for boxicity in terms of the maximum degree ∆ alone. This means that no matter how large a graph might be, a box representation in a small number of dimensions can be constructed for it if it has a small maximum degree. Most bounds on boxicity show that box(G) is small when the complement of G is small or sparse (for example, box(G) is bounded by the minimum size of a maximal matching in the complement; see [24]). This upper bound is perhaps the ﬁrst general bound showing that box(G) is small when G itself is small. We do not claim that this Chapter 2. Upper bounds for boxicity 25

upper bound is optimal; but make the following conjecture instead.

Conjecture. For any graph G with maximum degree ∆, box(G) is O(∆).

Roberts’ graphs are a family of graphs that have boxicity Ω(∆). In fact, we do not know of any graph that has boxicity greater than its maximum degree. Since box(G) n/2 when G has n vertices (as shown in [51]), the upper bound ≤ provided by Theorem 2.2 is of no use when ∆ > √n/2. Since for any graph G on n vertices with maximum degree ∆, box(G) (∆ + 2)ln n as shown by Theorem 2.1, ≤ ⌈ ⌉ the bound of 2∆2 given by Theorem 2.2 is better only when ∆ ln n. ≤ We are now armed with two upper bounds for the boxicity of general graphs in terms of the maximum degree. Both these bounds come in handy in the next chapter when we look at the boxicity of random graphs. As mentioned in Section 2.1.2, there are families of graphs for which the boxicity is exponentially larger than dav ln n, but we now exploit the power of probabilistic techniques to show that such graphs are rare.

Chapter 3

Boxicity of random graphs

Though an O(dav ln n) upper bound does not exist for boxicity of a general graph on n vertices with average degree dav, we now show that for almost all graphs, there does exist an upper bound for boxicity that is O(dav ln n). First, we shall look at some basics of the theory of random graphs.

3.1 Random graph preliminaries

Often, it is informative to look at graph properties from a statistical viewpoint. We could ask such questions as “if a graph is randomly drawn from a collection of graphs, what is the probability that the randomly chosen graph has property P ?”. In order to answer such questions, we need to deﬁne a probability space of graphs (we consider only ﬁnite graphs here) from which we draw a graph at random. The two most popularly used probability distributions (also called random graph models) are:

The (n, p) model: This is a probability space of all graphs on n vertices. The • G act of drawing a graph at random from this model is deﬁned by the following random experiment. Toss a coin that turns up heads with probability p for each n of the 2 possible edges. If the coin turns up heads, then we decide that the particular edge is present in the randomly drawn graph and the edge is not present otherwise. Thus, each edge has an independent probability of p of being present

27 Chapter 3. Boxicity of random graphs 28

in the randomly drawn graph. Clearly, this is not a uniform distribution over all graphs on n vertices. The probability of a graph with m edges to be the randomly n −m drawn graph is pm(1 p)(2) . Note that the distribution becomes uniform if − 1 p = 2 .

The (n, m) model: In this model, the randomly chosen graph is drawn uniformly • G at random from the collection of all graphs on n vertices with m edges. Thus the probability of any given graph on n vertices and m edges to occur as the randomly N n chosen graph is the same, i.e. 1/ m where N = 2 . We say that a given property P is true for almost all graphs if for a randomly chosen graph G from the random graph model under consideration, Pr[G has property P ] 1 → when n . This can be seen as the mathematical way of saying that the proportion → ∞ of graphs without property P becomes negligibly small as n becomes large and therefore “almost all” graphs can be thought to have this property.

3.2 Boxicity is O(dav ln n) for almost all graphs

The proof shows that for almost all graphs G drawn from the (n, m) model, box(G) G ∈ O(c ln n) where c = 2m/n (refer to Section 1.4 for a description of the asymptotic notation as we use it). We assume c > 1 as we are mainly interested in connected graphs. But we first show the result for the (n, p) model setting p = c/(n 1). As G − shown in [9], we can then carry over the result to the (n, m) model since p = m/ n . G 2 Consider the (n, p) model with p = c/(n 1). Let G denote a random graph drawn G − according to this model. For a vertex u, define a random variable du that denotes the degree of u, i.e. d = N(u) = e where e is an indicator random variable u | | v∈V (G),v6=u u,v u,v whose value is 1 if (u, v) E(GP) and 0 otherwise. Therefore, E[du]= p(n 1) = c. ∈ − Case 1: c ln n. ≥ Since du is the sum of independent Bernoulli random variables, we can use Chernoff bound to bound the probability of du becoming large. In particular, we use the following Chapter 3. Boxicity of random graphs 29

form of the Chernoﬀ bound given in [3] for the rest of the proof.

2E − δ [X] Pr[X (1 + δ)E[X]] e 2+δ (3.1) ≥ ≤ for all δ > 1. Taking δ = 5, we get, Pr[d 6c] 1/n3. Now, by the union u ≥ ≤ bound, it follows that Pr[∆(G) 6c] = Pr[ u V (G),d 6c] 1/n2. Using the re- ≥ ∃ ∈ u ≥ ≤ sult box(G) (∆ + 2)ln n , we now have, box(G) (6c + 2) ln n with probability at ≤ ⌈ ⌉ ≤ least 1 1/n2. − Case 2: c< ln n. Let S = V (G) N(u) u . u − − { } Let N ′(u)= v S u′ N(u) such that (u′,v) E(G) . { ∈ u | ∃ ∈ ∈ } In this case, we will use a diﬀerent technique to upper bound boxicity. Let the graph G2 denote the square of G. That is, V (G2)= V (G) and (u, v) E(G2) if there is a path ∈ of length 1 or 2 between u and v. Recall that the proof of Theorem 2.2 shows that for any graph G, box(G) 2χ(G2) 2∆(G2) + 2. We will show below that if c< ln n, then ≤ ≤ ∆(G2) c + 6ln n + 7c2 + 42c ln n, with high probability. The reader may note that ≤ the degree of a vertex u in G2 equals N(u) + N ′(u) . We will now show that for any | | | | vertex u, Pr[ N(u) + N ′(u) / O(c log n)] 3/n3. | | | | ∈ ≤ Let k = c +6ln n. We apply Chernoﬀ bound (3.1) with δ = 6ln n/c to obtain

Pr[d k] e−δ(6ln n)/(2+δ) 1/n3 u ≥ ≤ ≤

Let A V (G) such that A < k. Let Z(A) denote the event that N(u)= A. Now, for ⊆ | | each vertex v S , let X denote an indicator random variable indicating whether v ∈ u v,A ∈ N ′(u) conditioned on the event Z(A). Note that for any vertex v S , Pr[X = 1] ∈ u v,A ≤ kp. Let X = X . It follows that E[X ] kp(n 1) = kc. Since X is the sum A v∈Su v,A A ≤ − A of independentP Bernoulli random variables, we apply the Chernoﬀ bound (3.1) by ﬁxing δ = 6kc/E[X ] to obtain Pr[X 7kc] e−δ(6kc)/(2+δ) 1/n3. A A ≥ ≤ ≤ Chapter 3. Boxicity of random graphs 30

′ Let the random variable Xu denote the cardinality of N (u). We now have,

Pr[X 7kc d < k] = Pr[(X 7kc) Z(A)] u ≥ | u u ≥ ∧ A⊆V (XG),|A|

It follows that

Pr[X 7kc] = Pr[X 7kc d < k] Pr[d < k] u ≥ u ≥ | u u +Pr[X 7kc d k] Pr[d k] u ≥ | u ≥ u ≥ (1/n3)Pr[d < k]+(1/n3)Pr[X 7kc d k] 2/n3 ≤ u u ≥ | u ≥ ≤

Let t = N(u) + N ′(u) = d + X . Combining the bounds on the values of d and X , u | | | | u u u u we get, Pr[t k + 7kc] Pr[d k]+ Pr[X 7kc] 3/n3 u ≥ ≤ u ≥ u ≥ ≤

2 Observe that ∆(G ) = maxu∈G tu. Thus, by applying the union bound, we obtain

Pr ∆(G2) k + 7kc = Pr t k + 7kc 3/n2 ≥  u ≥  ≤ u∈V (G) _   Thus, with high probability, ∆(G2) < k + 7kc = c + 6ln n + 7c2 + 42c ln n. Recalling that box(G) 2∆(G2) + 2, we obtain box(G) O(c ln n) with high probability, since ≤ ∈ c< ln n. Having shown that in the (n, p) model, Pr[box(G) O(c ln n)] 3/n2, the following G 6∈ ≤ relation from page 35 of [9] helps us to extend our result to the (n, m) model. G

P (Q) 3m1/2P (Q) m ≤ p

where Q is a property of graphs of order n, and Pm(Q) and Pp(Q) are the probabilities of a graph chosen at random from the (n, m) or the (n, p) models respectively to have G G Chapter 3. Boxicity of random graphs 31

n property Q given that p = m/ 2 . Using this result, we now have, for a graph G drawn randomly from the (n, m) model, G

Pr[box(G) O(c ln n)] 9n−2√m 9/n 6∈ ≤ ≤

As c = 2m/n = dav, which is the average degree, we have shown that for almost all graphs with a given average degree dav, the boxicity is O(dav ln n). Thus we have the following theorem:

Theorem 3.1. For a random graph G on n vertices and m edges drawn according to (n, m) model, G 2m 9 Pr box(G) = O ln n 1 n ≥ − n

3.3 Remarks

We know that box(G) tw(G) + 2 [16]. It is well known that almost all graphs on n ≤ vertices and m = cn edges (for a suﬃciently large constant c) have treewidth Ω(n) [37]. From the discussion in this chapter, we know that almost all graphs on n vertices and m edges have boxicity O(dav ln n) where dav = 2m/n. An implication of this is that when c is a large enough constant, for almost all graphs on m = cn edges, there is an exponential gap between their boxicity and treewidth. Hence it is interesting to reconsider those NP- hard problems that are polynomial time solvable in bounded treewidth graphs and see whether they are also polynomial time solvable for bounded boxicity graphs.

Chapter 4

A randomized construction for cubicity

Let us now turn our attention to the cubicity of graphs. Recall that the cubicity of a graph is the minimum dimension in which it can be represented as the intersection of k-cubes. It is immediate that the cubicity of a graph is always at least its boxicity as a k-cube representation for a graph is also a k-box representation for it. It seems natural to think about the relationship between the boxicity and cubicity of a graph. Chandran and K. A. Mathew show in [15] that cub(G) log n for any graph G box(G) ≤ ⌈ 2 ⌉ on n vertices. In Chapter 2, we saw two upper bounds on the boxicity of any graph G on n vertices and having maximum degree ∆, namely, box(G)= O(∆ln n) and box(G) 2∆2. ≤ Combining these with the result cub(G) log n , we get cub(G) = O(∆ln2 n) and box(G) ≤ ⌈ 2 ⌉ cub(G) 2∆2 log n . In this chapter, we suitably adapt the randomized construction ≤ ⌈ 2 ⌉ of [18] to show that cub(G) is O(∆ln n) which is an improvement over both these bounds on cubicity. Let G be a graph on n vertices with maximum degree ∆. We ﬁrst show a randomized algorithm RAND to construct the cube representation of G in 4(∆ + 1) ln n ⌈ ⌉ dimensions. We then give a detailed exposition of the derandomization technique by demonstrating how the algorithm RAND can be derandomized to obtain a polynomial time deterministic algorithm DET that gives a cube representation of G in the same

33 Chapter 4. A randomized construction for cubicity 34

number of dimensions. Both these algorithms compute an indifference graph representation of G using 4(∆ + 1) ln n indifference graphs. The algorithms construct equal ⌈ ⌉ interval representations (recall the definition from Section 1.3) for each graph in the indifference graph representation.

4.1 The algorithm RAND

In this section we describe the randomized algorithm RAND that computes a cube representation in O(∆ln n) dimensions for any graph G on n vertices and maximum degree ∆ . For ease of notation we will let V = V (G) for the remainder of this chapter. The reader might find it useful to recall the definition of a permutation as given in Definition 1.2.

Deﬁnition 4.1. Let π be a permutation of a set S. Let X S. The restriction of π ⊆ onto X, denoted as π , is a permutation of X deﬁned as follows. Let X = u ,...,u X { 1 r} such that π(u ) <π(u ) < <π(u ). Then π (u ) = 1, π (u ) = 2,...,π (u )= r. 1 2 ··· r X 1 X 2 X r

Construction of the indifference supergraph (G, π, A): M Let π be a permutation on V and let A be a subset of V . We define (G, π, A) to be M an indifference graph G′ with V (G′)= V constructed as follows. Let B = V A. We shall construct f, an equal interval representation (recall − Definition 1.16) with interval length n for G′ as follows: u B, define f(u)= n + π(u), ∀ ∈ u A and N(u) B = , define f(u) = 0, ∀ ∈ ∩ ∅ u A and N(u) B = , define f(u) = max π(x). ∀ ∈ ∩ 6 ∅ x∈N(u)∩B Thus, two vertices u and v will have an edge in G′ if and only if f(u) f(v) n. | − | ≤ Clearly, G′ is an indifference graph. It can be seen that the vertices in B induce a clique in G′ as the intervals assigned to each of them contain the point 2n. Similarly, all the Chapter 4. A randomized construction for cubicity 35

vertices in A also induce a clique in G′ as the intervals mapped to each contain the point n. Now, we show that G′ is a supergraph of G. To see this, take any edge (u, v) E(G). ∈ If u and v both belong to A or if both belong to B, then (u, v) E(G′) as we have ∈ observed above. If this is not the case, then we can assume without loss of generality that u A and v B. Let t = max π(x). Obviously, t π(v), since v N(u) B. ∈ ∈ x∈N(u)∩B ≥ ∈ ∩ From the deﬁnition of f, we have f(u) = t and we have f(v) = n + π(v). Therefore, f(v) f(u) = n + π(v) t and since t π(v), it follows that f(v) f(u) n. This − − ≥ − ≤ shows that (u, v) E(G′). ∈

We are now ready to give the randomized algorithm RAND that, given an input graph G, outputs an indiﬀerence supergraph G′ of G.

RAND Input: G. Output: G′ which is an indiﬀerence supergraph of G. begin 1. Generate a permutation π of V uniformly at random. 2. for each vertex u V , ∈ Toss an unbiased coin to decide whether u should belong to A or to B (i.e. Pr[u A]= Pr[u B]= 1 ). ∈ ∈ 2 3. return G′ = (G, π, A). M end

Lemma 4.2. Let e = (u, v) / E(G). Let G′ be the graph returned by RAND(G). ∈ Then,

1 1 d(u) d(v) Pr[e E(G′)] + + ∈ ≤ 2 4 d(u) + 1 d(v) + 1 2∆ + 1 ≤ 2∆ + 2 where d(u) and d(v) denote the degrees of the vertices u and v respectively in G. Chapter 4. A randomized construction for cubicity 36

Proof: Let π be the permutation and A, B be the partition of V generated randomly { } by RAND(G). An edge e =(u, v) / E(G) will be present in G′ if and only if one of the ∈ following cases occur:

1. Both u, v A or both u, v B ∈ ∈ 2. u A, v B and max π(x) >π(v) ∈ ∈ x∈N(u)∩B 3. u B,v A and max π(x) >π(u) ∈ ∈ x∈N(v)∩B

Let P1 denote the probability of situation 1 to occur, P2 that of situation 2 and P3 that of situation 3. Since all the three cases are mutually exclusive, Pr[e E(G′)] = P +P +P . ∈ 1 2 3 It can be easily seen that P = Pr[u, v A]+ Pr[u, v B]= 1 + 1 = 1 . P and P can 1 ∈ ∈ 4 4 2 2 3 be calculated as follows:

P2 = Pr u A v B max π(x) >π(v) ∈ ∧ ∈ ∧ x∈N(u)∩B

Note that creating the random permutation and tossing the coins are two diﬀerent ex- periments independent of each other. Moreover, the coin toss for each vertex is an experiment independent of all other coin tosses. Thus, the events u A, v B and ∈ ∈ maxx∈N(u)∩B π(x) >π(v) are all independent of each other. Therefore,

P2 = Pr[u A] Pr[v B] Pr max π(x) >π(v) ∈ × ∈ × x∈N(u)∩B

Now, Pr max π(x) >π(v) Pr max π(x) >π(v) = p (say). Let X = x∈N(u)∩B ≤ x∈N(u) v N(u) and let πX be the restriction of π onto X. Then p is the probability { } ∪ that the condition π (v) = X is satisﬁed. Since π can be any permutation of X 6 | | X X = d(u) + 1 elements with equal probability 1 and the number of permu- | | (d(u)+1)! d(u)!d(u) d(u) tations which satisfy our condition is d(u)!d(u), p = (d(u)+1)! = d(u)+1 . Therefore, Pr max π(x) >π(v) d(u) . It can be easily seen that Pr[u A] = 1 and x∈N(u)∩B ≤ d(u)+1 ∈ 2 Pr[v B]= 1 . Thus, ∈ 2 1 1 d(u) 1 d(u) P = 2 ≤ 2 × 2 × d(u) + 1 4 d(u) + 1 Chapter 4. A randomized construction for cubicity 37

Using similar arguments, 1 d(v) P 3 ≤ 4 d(v) + 1 Thus,

Pr[e E(G′)] = P + P + P ∈ 1 2 3 1 1 d(u) d(v) + + ≤ 2 4 d(u) + 1 d(v) + 1

Hence the lemma.

Theorem 4.3. Given a simple, undirected graph G on n vertices with maximum degree ∆, cub(G) 4(∆ + 1) ln n . ≤ ⌈ ⌉ Proof: Let us invoke RAND(G) k times so that we obtain k indifference supergraphs of G which we will call G′ ,G′ ,...,G′ . Let G′′ = G′ G′ G′ . Obviously, G′′ is a 1 2 k 1 ∩ 2 ∩···∩ k supergraph of G. If G′′ = G, then we have obtained an indifference graph representation for G using k indifference graphs, which means that cub(G) k. We now estimate an ≤ upper bound for the value of k so that G′′ = G. Let (u, v) / E(G). ∈

′′ ′ Pr[(u, v) E(G )] = Pr (u, v) E(Gi) ∈ " ∈ # 1≤^i≤k 2∆ + 1 k (From Lemma 4.2) ≤ 2∆ + 2 Chapter 4. A randomized construction for cubicity 38

Pr[G′′ = G] = Pr (u, v) E(G′′) 6  ∈  (u,v)_∈/E(G) n2  2∆ + 1 k  ≤ 2 2∆ + 2 n2 1 k = 1 2 − 2(∆ + 1) 2 n − k e 2(∆+1) ≤ 2 ×

Note that we used the inequality 1 + x ex for the last step of the derivation. Now, ≤ choosing k = 4(∆ + 1) ln n, we get,

1 Pr[G′′ = G] 6 ≤ 2

Therefore, if we invoke RAND k = 4(∆+1) ln n times, there is a non-zero probability ⌈ ⌉ ′ ′ ′ that G1,G2 ...,Gk form an indiﬀerence graph representation of G. Thus, there exists an indiﬀerence graph representation of G using 4(∆ + 1) ln n graphs which implies that ⌈ ⌉ cub(G) 4(∆ + 1) ln n . ≤ ⌈ ⌉

Theorem 4.4. Given a graph G on n vertices with maximum degree ∆. Let G1,G2,...,

Gk be k indiﬀerence supergraphs of G generated by k invocations of RAND(G) and let G′′ = G′ G′ . . . G′ . Then, for k 6(∆ + 1) ln n, G′′ = G with high probability. 1 ∩ 2 ∩ ∩ k ≥ Proof: Choosing k = 6(∆ + 1) ln n in the ﬁnal step of proof of Theorem 4.3, we get,

1 Pr[G′′ = G] 6 ≤ 2n

Thus, if k 6(∆ + 1) ln n, G′′ = G with high probability. ≥

Theorem 4.5. Given a graph G with n vertices, m edges and maximum degree ∆, with high probability, its cube representation in 6(∆ + 1) ln n dimensions can be generated ⌈ ⌉ Chapter 4. A randomized construction for cubicity 39

in O(∆(m + n) ln n) time. Proof: We assume that a random permutation π on n vertices can be computed in O(n) time and that a random coin toss for each vertex takes only O(1) time. We take n steps to assign intervals to the n vertices. Suppose in a given step, we are attempting to assign an interval to vertex u. If u B, then we can assign the interval [n + π(u), 2n + π(u)] ∈ to it in constant time. If u A, we look at each neighbour of the vertex u in order to ∈ ﬁnd out a neighbour v B such that π(v) = max π(x) and assign the interval ∈ x∈N(u)∩B [π(v), n+π(v)] to u. It is obvious that determining this neighbour v will take just O(d(u))

1 time. Since the number of edges in the graph m = 2 Σu∈V d(u), one invocation of RAND needs only O(m + n) time. Since we need to invoke RAND O(∆ln n) times (see the proof of Theorem 4.3), the overall algorithm that generates the cube representation in 6(∆ + 1) ln n dimensions runs in O(∆(m + n) ln n) time. ⌈ ⌉

4.2 Derandomizing RAND

The above algorithm can be derandomized by adapting the techniques used in [18] to obtain a deterministic polynomial time algorithm DET with the same performance guar- antee on the number of dimensions for the cube representation. Let t = 4(∆ +1) ln n . Given G, DET selects t permutations π ,...,π and t subsets ⌈ ⌉ 1 t A ,...,A of V in such a way that the indiﬀerence graphs (G, π ,A ) 1 i t 1 t {M i i | ≤ ≤ } form an indiﬀerence graph representation of G.

4.2.1 Some notations

A permutation π can also be written as an ordered set of vertices v ,v ,...,v . This h 1 2 ni notation means that π−1(i) = v , for 1 i n. Let b : V 0, 1 so that b(v) = 0 i ≤ ≤ → { } denotes v A and b(v) = 1 denotes v B. We construct π by choosing the vertices ∈ ∈ v1,v2,...,vn in that order. As we choose each vertex v, we also decide whether it should belong to A or B by setting the bit b(v) to 0 or 1. After step i, we have an ordered set Chapter 4. A randomized construction for cubicity 40

of i “vertex-bit” pairs, V = (v , b ), (v , b ),..., (v , b ) where b = b(v ), for 1 j i. i h 1 1 2 2 i i i j j ≤ ≤ Let Vˆ = v ,v ,...,v . Also define function m : Vˆ 0, 1 where m (v ) = b , for i { 1 2 i} Vi i → { } Vi j j 1 j i. Let π : Vˆ 1,...,i denote the ordering of Vˆ defined by π (v ) = j. ≤ ≤ Vi i → { } i Vi j Note that π can also be seen as a permutation of V . Also let A = v : m (v ) = 0 . Vn Vi { j Vi j } We also define an operator as: ⋄ V (u, c)= (v , b ), (v .b ),..., (v , b ), (u, c) . i ⋄ h 1 1 2 2 i i i

4.2.2 A closer look at RAND

Observe that in RAND, G′ is the outcome of a random experiment since in essence, RAND computes a random permutation π and selects the bit b(v) (mentioned above) for each vertex v at random. For each non-edge e = (u, v) E(G), define a random ∈ variable x such that x = 0 if and only if one of the following is true : (i) Both u, v A e e ∈ or both u, v B, (ii) u A, v B and max π(x) > π(v), (iii) u B,v A and ∈ ∈ ∈ x∈N(u) ∈ ∈ maxx∈N(v) π(x) >π(u). We set xe = 1 for all other cases. It can be easily observed that (x = 1) e E(G′). e ⇒ 6∈ For any set H E(G), define random variable X = x . It is easy to see that ⊆ H e∈H e ′ there will be at least XH edges in H that are missing in GP. Given V = (v , b ), (v , b ),..., (v , b ) , let C(V ) denote the event that v ,v ,...,v i h 1 1 2 2 i i i i 1 2 i form the first i elements of the permutation π and b(v )= b for 1 j i. j j ≤ ≤ Define f (V )= E[x C(V )] = Pr[x = 1 C(V )]. Also for H E(G), define F (V )= e i e| i e | i ⊆ H i E[X C(V )] = f (V ). H | i e∈H e i We will let VP0 denote the empty ordering—i.e., one that contains no vertex-bit pairs.

Thus, C(V0) is the event that the status of every vertex (meaning the position in the ﬁnal permutation and whether the vertex should belong to set A or B) is undetermined. Therefore, f (V ) = Pr[x = 1] 1 (note that the proof of Lemma 4.2 actually e 0 e ≥ 2(∆+1) proves that Pr[x = 0] 2∆+1 ) and therefore F (V )= f (V ) |H| . e ≤ 2∆+2 H 0 e∈H e 0 ≥ 2(∆+1) P Chapter 4. A randomized construction for cubicity 41

4.2.3 Constructing the permutations and subsets

Given H E(G), we deterministically construct a permutation π and a subset A of V ⊆ so that at least |H| of the non-edges in H are missing in (G, π, A). Our strategy is 2(∆+1) M to start with V0 and construct V1, V2,..., Vn in n steps. The final permutation π and the subset A of V are given by πVn and AVn respectively. After step i, we have determined an ordering Vi of vertex-bit pairs. During step i + 1, we find a suitable vertex-bit pair (u, c) where u V Vˆ and c 0, 1 that can be added to V using the operator so as ∈ − i ∈ { } i ⋄ to get V . Recall that F (V ) is actually E[X V ], i.e., it is the expected value of X i+1 H i H | i H if in the ith step, we have determined the status of i vertices as given in Vi. When we are constructing V in the the (i + 1)th step, we have 2 V Vˆ possible choices, since i+1 | − i| we can pick any of the V Vˆ remaining vertices to be u and at the same time we have | − i| two choices for c—0 or 1. Thus, after any step i, we have 2 V Vˆ possible choices for | − i| V . It can be easily seen that E[X V ] is the average of E[X V ] values over all the i+1 H | i H | i+1 different choices of Vi+1. Therefore,

F (V ) = E[X V ] H i H | i 1 = F (V (u, c)) ˆ  H i ⋄  2 V Vi ˆ | − | u∈V −XVi,c∈{0,1}  

Now, in order to construct Vi+1, we take such a vertex as u and such a value for c and make V = V (u, c) so that F (V ) is maximized (we shall show later that F (V ) i+1 i ⋄ H i+1 H i+1 can be calculated in polynomial time). It is obvious that if we proceed in this manner, F (V ) F (V ), for 0 i < n. Therefore, F (V ) F (V ) |H| . Note that H i+1 ≥ H i ≤ H n ≥ H 0 ≥ 2(∆+1) if G′ = (G, π, A), where π = π and A = A , then F (V ) H E(G′) . Thus, M Vn Vn H n ≤ | ∩ | H E(G′) |H| . We can summarize the procedure for constructing V and the | ∩ | ≥ 2(∆+1) n ′ ′ indiﬀerence graph G associated with Vn as the algorithm DET given below:

DET′ Input: G, H E(G). ⊆ Output: G′ which is an indiﬀerence supergraph of G, such that Chapter 4. A randomized construction for cubicity 42

E(G′) H |H| . | ∩ | ≥ 2(∆+1) begin for i from 1 to n max := 0, v := (0, 0) for u V V ∈ − i−1 for c 0, 1 ∈ { } f := F (V (u, c)) H i−1 ⋄ if f max, then v := (u, c), max := f ≥ V := V v i i−1 ⋄ return G′ = (G, π ,A ). M Vn Vn end

It is easily observed that DET′ runs deterministically in polynomial time if each

FH (Vi) can be computed in polynomial time. But calculation of each FH (Vi)

= e∈H fe(Vi) in polynomial time is possible only if we can calculate fe(Vi) in poly- nomialP time.

4.2.4 Calculating fe(Vi)

Let e = (u, v) E(G). f (V ) is the probability that x = 1 given C(V ) has happened. ∈ e i e i We will analyze the diﬀerent situations that can occur. We will let π denote the permutation given by πVn .

1. If u, v Vˆ : ∈ i In this case, the status of u and v have already been determined. Therefore, we can tell for sure whether x is 1 or 0. Recalling that f (V ) = Pr[x = 1 C(V )], e e i e | i this means that fe(Vi) will be either 1 or 0. If the bits that have been selected for

u and v, mVi (u) and mVi (v) respectively, are equal, then u and v are either both in

A or both in B. In that case, xe = 0 and therefore fe(Vi) = 0. Now, consider the Chapter 4. A randomized construction for cubicity 43

case m (u) = m (v). Let us assume without loss of generality that u A and Vi 6 Vi ∈ v B. If N(u) Vˆ , there is some neighbour x of u such that π(x) >π(v). Even ∈ 6⊆ i if N(u) Vˆ , there may be some neighbour of u, say x, such that π (x) >π (v). ⊆ i Vi Vi In both these cases, x = 0 by deﬁnition of x . Thus x = 1 only if N(u) Vˆ and e e e ⊆ i

maxx∈N(u) πVi (x) <πVi (v). We summarize these below:

Case 1 : If mVi (u)= mVi (v), fe(Vi) = 0 Case 2 : If m (u) = m (v) Vi 6 Vi let u A, v B. ∈ ∈ Case 2.1 : If N(u) Vˆ and max π (x) <π (v), f (V ) = 1 ⊆ i x∈N(u) Vi Vi e i Case 2.2 : otherwise, fe(Vi) = 0

2. If u Vˆ ,v Vˆ : ∈ i 6∈ i Here, we know about u’s position in the ﬁnal permutation and also whether u is in set A or B. But we have no such information about v.

If u A, then x = 1 if and only if v B and also max π(x) < π(v). This ∈ e ∈ x∈N(u) means that all neighbours of u should come before v in the ﬁnal permuation π.

We know that those neighbours of u that are in Vˆi will anyway come before v in

the final permutation. Now, let Mu denote the set of neighbours of u that are not there in Vˆ , i.e., M = N(u) (V Vˆ ). Let k = M . It is easy to see that i u ∩ − i u | u| f (V ) is the probability that v B and all the vertices in M come before v in the e i ∈ u final permutation. Obviously, Pr[v B] = 1 . Now, let X = M v . Consider ∈ 2 u ∪ { } the restriction of π onto X, denoted by π . Pr max π(x) <π(v) C(V ) = X x∈N(u) | i Pr[v is the last element in the permutation π ]= Pr[π (v)= k +1] = ku! = X X u (ku+1)! 1 . Therefore, f (V )= 1 . ku+1 e i 2(ku+1) If u B, then f (V ) is the probability that v A and all neighbours of v come ∈ e i ∈ before u in the final permutation. Whether all neighbours of v come before u can be determined right away as we have already created the permutation at least till the position of u. Thus we check whether N(v) Vˆ and max π (x) <π (u) ⊆ i x∈N(v) Vi Vi Chapter 4. A randomized construction for cubicity 44

and set fe(Vi) = 0 if not. If the condition is satisﬁed, still v might be put in set B 1 1 itself with probability 2 and thus xe can become 0. Thus, we set fe(Vi)= 2 if the condition is satisﬁed.

We summarize below: Case 1 : If u A (i.e. m (u) = 0), ∈ Vi

1 fe(Vi) = , where ku = N(u) (V Vˆi) . 2(ku + 1) | ∩ − |

Case 2 : otherwise, (i.e. u B,m (u)=1) ∈ Vi Case 2.1 : if N(v) Vˆ and max π (x) <π (u), f (V )= 1 ⊆ i x∈N(v) Vi Vi e i 2 Case 2.2 : otherwise, fe(Vi) = 0

3. If u, v Vˆ : 6∈ i

The positions of neither u nor v have been determined. fe(Vi) is the probability

that xe = 1, which is the probability that given C(Vi) has happened, (i) u A and v B and max π(x) <π(v), or ∈ ∈ x∈N(u) (ii) u B and v A and max π(x) <π(u). ∈ ∈ x∈N(v) Note that cases (i) and (ii) are mutually exclusive. Let Mu denote the set of neighbours of u that are not present in Vˆ , i.e. M = N(u) (V Vˆ ). Similarly, let M = i u ∩ − i v N(v) (V Vˆ ). Let k = M and k = M . As we observed in the previous sec- ∩ − i u | u| v | v| tion, Pr max π(x) <π(v) = 1 . Similarly, Pr max π(x) <π(u) = x∈N(u) ku+1 x∈N(v) 1 . Also, it is easy to see that Pr[u A v B] = Pr[u B v A] = 1 . kv+1 ∈ ∧ ∈ ∈ ∧ ∈ 4 Chapter 4. A randomized construction for cubicity 45

Therefore, fe(Vi) can be computed as,

fe(Vi) = Pr[u A v B] Pr max π(x) <π(v) ∈ ∧ ∈ × x∈N(u) +Pr[u B v A] Pr max π(x) <π(u) ∈ ∧ ∈ × x∈N(v) 1 1 1 1 = + 4 k + 1 4 k + 1 u v 1 1 1 = + 4 k + 1 k + 1 u v

Searching for a given vertex in the set Vˆi obviously takes only polynomial time. Since the neighbours of any given vertex can also be determined in polynomial time, it follows that the value ku for any vertex u can be computed in polynomial time as well.

Therefore, at any given stage, fe(Vi) and hence FH (Vi) can be computed in polynomial time. Thus, it follows that the algorithm DET′ runs in polynomial time. All of this can be summarized in the following lemma.

Lemma 4.6. DET′, on input G and H, where H E(G), outputs in polynomial time ⊆ an indiﬀerence supergraph G′ of G such that E(G′) H |H| . | ∩ | ≥ 2(∆+1) Proof: Follows from the discussions in the previous sections.

4.2.5 The algorithm DET

Our main algorithm DET constructs the indifference graph representation of an input graph G using 4(∆ + 1) ln n indifference graphs. It invokes DET′ as a subroutine. It ⌈ ⌉ initially sets H to be the set of non-edges in G and runs DET′ with G and H as input. The indifference graph G′ output by DET′ will have some non-edges in H missing. We remove those non-edges from H and repeat the procedure. Each time, G′ is added to a list L of indifference supergraphs of G. The algorithm stops when H becomes empty, i.e. every non-edge in G is missing in one of the indifference graphs that have been added to L. The algorithm then outputs L as the indifference graph representation of G. Chapter 4. A randomized construction for cubicity 46

DET Input: G. Output: An indiﬀerence graph representation of G. begin L := ∅ H := E(G) while H = 6 ∅ G′ := DET′(G, H) Add G′ to L H := H E(G′) − return L. end

Let Hi denote the set H after i iterations of the while loop. Therefore, H0 = E(G). From Lemma 4.6, we have

1 i H H 1 | i| ≤ | 0| − 2(∆ + 1) n2 1 i 1 ≤ 2 − 2(∆ + 1) 2 n −i e 2(∆+1) . ≤ 2 ·

For i 4(∆ + 1) ln n, H 1/2 < 1. Therefore H becomes empty after 4(∆ + 1) ln n ≥ | i| ≤ ⌈ ⌉ iterations implying that the while loop does not run for more than that many iterations. Each graph added to L is an indifference supergraph of G and each non-edge in H is removed only when that non-edge is missing in the graph just added to L. Thus, when the loop exits, L is a set of indifference supergraphs of G such that each non-edge in G is missing in at least one graph in L. This shows that DET outputs an indifference graph representation using 4(∆ + 1) ln n graphs. ⌈ ⌉ Chapter 4. A randomized construction for cubicity 47

Tight example:

Consider the case when G is a complete binary tree of height d = log n. Using the

d log n results shown in [14], we can see that cub(G) = where c1 is a constant. ≥ log 2d c1+log log n Therefore, cub(G)=Ω( log n ). From Theorem 4.3, cub(G) 4(∆ + 1) ln n = 16 ln n = log log n ≤ c2 log n, where c2 is a constant. Therefore, the upper bound provided by Theorem 4.3 is tight up to a factor of O(log log n).

4.3 A useful result

The simple technique of randomly constructing indiﬀerence supergraphs of a graph has helped us prove that for any graph G on n vertices and having maximum degree ∆, cub(G) 4(∆ + 1) ln n . We had seen in Chapter 2 that box(G) = O(∆ln n). Now ≤ ⌈ ⌉ we know that even cub(G)= O(∆ln n). Section 4.2 showed how the randomness in the procedure can be removed to obtain a deterministic algorithm that constructs the cube representation of an input graph in 4(∆ + 1) ln n dimensions. ⌈ ⌉ What makes this upper bound more interesting is the fact that it comes in handy while proving various other results about cubicity. In the next chapter, the upper bound and the construction used to derive it are employed to prove a new upper bound on cubicity in terms of the bandwidth of the graph. The same construction is used again in Chapter 6 where we show that the upper bound of O(∆ln n) on cubicity can be improved substantially for the class of interval graphs.

Chapter 5

Cubicity and bandwidth

Given an undirected graph G =(V,E) on n vertices, a linear ordering of G is a bijection f : V (G) 1,...,n . The width of the linear ordering f is deﬁned as max f(u) → { } (u,v)∈E | − f(v) . The bandwidth minimization problem is to compute f with minimum possible | width.

Deﬁnition 5.1. The bandwidth of G denoted as bw(G) is the minimum possible width achieved by any linear ordering of G. A bandwidth ordering of G is a linear ordering of G with width bw(G).

It can be easily seen that if ∆ is the maximum degree of G, then ∆/2 bw(G) ⌈ ⌉ ≤ ≤ n 1. We now present two upper bounds on the cubicity of a graph in terms of its − bandwidth. For any graph G with bandwidth b and maximum degree ∆,

cub(G)= O(∆ln b) • We make use of the construction used in the proof of Theorem 4.3 and improve the O(∆ln n) bound given by the theorem. A deterministic algorithm that outputs the cube representation of a graph in O(∆ln b) dimensions given a bandwidth ordering of it is presented. Note that the bandwidth b is at most n and b is much smaller than n for many well-known graph classes.

cub(G) b + 1 • ≤ We analyze the bandwidth ordering of a graph in detail and show that there exists

49 Chapter 5. Cubicity and bandwidth 50

a cube representation in b + 1 dimensions for any graph with bandwidth b. The proof can be used to construct a deterministic algorithm that outputs the cube representation of an input graph G in b + 1 dimensions, given a linear ordering of G with width b in O(b n) time. Note that in cases where ∆ is Ω(b/ log b), this · algorithm produces a cube representation in a lower number of dimensions than the previous one.

Combining the above two algorithms we can construct the cube representation of G in O(min b, ∆ln(b) ) dimensions given a linear ordering of G with width b in polynomial { } time. Clearly, this upper bound on cubicity is exponentially better than Roberts’ bound of 2n/3 [51] for many well-known graph classes.

A note on bandwidth computation:

Our algorithms to compute the cube representation of a graph G take as input a linear ordering of G. The smaller the width of this ordering, the lesser the number of dimensions of the cube representation of G computed by these algorithms. Thus, it would be best if a bandwidth ordering of G can be obtained. But computing the bandwidth is an NP- complete problem and approximating the bandwidth of G within a ratio better than k for every k N is also NP-complete [61]. Feige [29] gives a O(log3(n)√log n log log n) ∈ factor approximation algorithm to compute the bandwidth and also the corresponding linear ordering for general graphs. We can use this algorithm in combination with our ﬁrst algorithm to obtain a polynomial time deterministic algorithm to construct the cube representation of G in O(∆(ln b+ln ln n)) dimensions, given only G. Also, for bandwidth computation, several algorithms with good heuristics are known that perform very well in practice [60].

5.1 Cube representation in O(∆ ln b) dimensions

In this section we show an algorithm DETBAND to construct the cube representation of G =(V,E) in O(∆ln b) dimensions given a linear arrangement of V (G) with width A Chapter 5. Cubicity and bandwidth 51

b. The DETBAND algorithm internally invokes the DET algorithm (see Section 4.2). Let the linear arrangement be v ,...,v . For ease of presentation, assume that n is A 1 n a multiple of b. Define a partition B0,...,Bk−1 of V (G) where k = n/b, where Bj = v ,...,v . Let H for 0 i k 2 be the induced subgraph of G on the vertex set { jb+1 jb+b} i ≤ ≤ − B B . Since for any i, V (H ) = 2b, we have cub(H ) 4(∆ + 1) ln(2b) = t (say). i ∪ i+1 | i | i ≤ ⌈ ⌉ 1 t Let Hi ,...,Hi be the indifference graph representation for Hi produced by DET when 1 t given Hi as the input. Let gi ,...,gi be their corresponding equal interval representations with interval length n 1. We shall define graphs I , G , G and G such that G = I G G G . Clearly, 0 0 1 2 0 ∩ 0 ∩ 1 ∩ 2 these graphs all need to be supergraphs of G such that any edge not present in G is missing in at least one of them. We can categorize the non-edges in G into the following classes:

1. (u, v) E(G) such that u B and v B and i j > 1, 6∈ ∈ i ∈ j | − | 2. (u, v) E(G) such that u B and v B , and 6∈ ∈ i ∈ i+1 3. (u, v) E(G) such that u, v B . 6∈ ∈ i

We construct I0 in such a way that all non-edges of type 1 are missing in I0. For 0 s k 2, the non-edges between vertices in H (which includes all the non-edges ≤ ≤ − s between blocks Bs and Bs+1 and also the type 3 non-edges in Bs and Bs+1) are taken care of in the graph Gs mod 3. Note that the type 3 non-edges in Bs will be missing in both Gs mod 3 and G(s−1) mod 3. The formal definition of these graphs follows. Define, for 0 i 2, the graph G with V (G ) = V (G) as the intersection of t ≤ ≤ i i indifference graphs Ii,1,...,Ii,t. The indifference graph Ii,j is defined by fi,j, an equal interval representation with interval length n for it. For each vertex u V (G ), define ∈ i fi,j(u) as follows: If u V (H ) such that s i,i + 3,i + 6,... , then define f (u)= gj(u). ∈ s ∈ { } i,j s Otherwise, define fi,j(u)= n.

1Note that throughout this chapter, the term “equal interval representation” is considered to be defined in the way it is defined in Definition 1.16. Chapter 5. Cubicity and bandwidth 52

The indifference graph I0 is constructed by assigning to each vertex in Bi the interval [in, (i + 1)n], for 0 i k 1. ≤ ≤ − We prove that G = I G G G which by Corollary 1.20 shows that cub(G) 0 ∩ 0 ∩ 1 ∩ 2 ≤ 3t + 1 12(∆ + 1) ln(2b) + 1 or cub(G)= O(∆ln b). ≤ ⌈ ⌉ The construction described above is given below as the algorithm DETBAND that given G and an arrangement with width b of V (G), outputs I I 0 i A { 0} ∪ { i,j | ≤ ≤ 2 and 1 j t , an indifference graph representation of G using 3t + 1 indifference ≤ ≤ } graphs where t = 4(∆+1) ln(2b) . In fact, DETBAND outputs equal interval represen- ⌈ ⌉ tations with interval length n for each graph in the indifference graph representation—f0 for I0 and fi,j for each graph Ii,j.

Deﬁnition 5.2. Let V and V be disjoint sets and let f : V R and f : V R 1 2 1 1 → 2 2 → be two functions. The union of f and f is the function f : V V R deﬁned as 1 2 1 ∪ 2 → follows:

f1(u), if u V1 and f(u)= ∈  f (u), if u V .  2 ∈ 2  Let t = 4(∆ + 1) ln(2b) . ⌈ ⌉

DETBAND Input: G, . A Output: The indifference graph representation I I 0 i 2 and { 0} ∪ { i,j | ≤ ≤ 1 j t of G using 3t + 1 indifference graphs. ≤ ≤ } begin Construction of I : for each i and for each node v B , f (v)= i n. 0 ∈ i 0 · Construction of I , 0 i 2 and 1 j t: i,j ≤ ≤ ≤ ≤ for 0 i 2, ≤ ≤ Invoke DET on each induced subgraph in = H : r = 0, 1,... . H { 3r+i } 1 t Let Hk ,...,Hk be the indifference graphs output by DET for Hk. l Let gk denote the equal interval representation with interval length n Chapter 5. Cubicity and bandwidth 53

l that DET produces for Hk. Let S = V (G) V (H). − H∈H R Let fS : S beS deﬁned as fS(v)= n for all v S. → ∈ for 1 j t, deﬁne f as the union of f and the functions ≤ ≤ i,j S in gj : r = 0, 1,... . { 3r+i } end

Theorem 5.3. DETBAND constructs the cube representation of G in at most 12(∆+ 1) ln(2b) + 1 dimensions in polynomial time. ⌈ ⌉ Proof: Let t = 4(∆+1) ln(2b) and let be v ,v ,...,v . Note that if (v ,v ) E(G), ⌈ ⌉ A 1 2 n x y ∈ then x y b since has width b. | − | ≤ A Claim 1. I0 is a supergraph of G. Proof: Consider an edge (v ,v ) E(G) (assume x < y). If B is the block containing x y ∈ m v , then v is contained in either B or B since y x b and each block contains b x y m m+1 − ≤ vertices. Thus, f (v )= mn and f (v )= mn or mn+n. In either case, f (v ) f (v ) 0 x 0 y | 0 x − 0 y | ≤ n and therefore, (v ,v ) E(I ). x y ∈ 0

Claim 2. I , for 0 i 2 and 1 j t, is a supergraph of G. i,j ≤ ≤ ≤ ≤ Proof: Consider an edge (v ,v ) E(G) (assume x < y). Let B be the block that x y ∈ m contains vx. As we have seen earlier, vy is either in Bm or in Bm+1. We shall show that

(vx,vy) is an edge in the indifference graph Ii,j. First, we make the following observation. If v ,v V (H ), where p = 3r + i for x y ∈ p some r 0, then by definition of f , f (v ) = gj(v ) and f (v ) = gj(v ), where gj ≥ i,j i,j x p x i,j y p y p is the equal interval representation with interval length n constructed by DET for the indifference graph Hj. Since (v ,v ) E(H ) and E(H ) E(Hj), gj(v ) gj(v ) n p x y ∈ p p ⊆ p | p x − p y | ≤ implying that f (v ) f (v ) n. Therefore (v ,v ) E(I ). | i,j x − i,j y | ≤ x y ∈ i,j Now, if m = 3r + i, for some r 0, then since v ,v H , it follows from the ≥ x y ∈ m discussion in the previous paragraph that (v ,v ) E(I ). x y ∈ i,j If m = 3r + i + 1, for some r 0, then we look at the following two cases: either ≥ v B or v B . In the first case, we have v ,v V (H ) and therefore the y ∈ m y ∈ m+1 x y ∈ m−1 Chapter 5. Cubicity and bandwidth 54

earlier argument can be applied again to obtain the result that(v ,v ) E(I ). Now, if x y ∈ i,j v B , we have v V (H ) and v S. Since m 1 = 3r +i, by deﬁnition of f , y ∈ m+1 x ∈ m−1 y ∈ − i,j f (v ) = gj (v ). From the construction of DET, it is clear that 0 f (v ) 2n. i,j x m−1 x ≤ i,j x ≤ Also, we have f (v ) = f (v ) = n. Therefore, it follows that f (v ) f (v ) n i,j y S y | i,j x − i,j y | ≤ and therefore (v ,v ) E(I ). x y ∈ i,j Similarly, if m = 3r + i + 2, for some r 0, then v S and v is contained either ≥ x ∈ y in S or in V (Hm+1) depending on whether vy is in Bm or Bm+1. It can be shown using arguments similar to the ones used in the preceding paragraph that (v ,v ) E(I ). x y ∈ i,j This completes the proof that E(G) E(I ), for 0 i 2, 1 j t. ⊆ i,j ≤ ≤ ≤ ≤

Claim 3. The indifference graphs I , for 0 i 2 and 1 j t, along with I i,j ≤ ≤ ≤ ≤ 0 constitute a valid indifference graph representation of G. Proof: We have to show that given any non-edge (v ,v ) E(G), there is at least one x y 6∈ graph among the 3t + 1 indifference graphs generated by DETBAND that does not contain the edge (vx,vy).

Assume that x < y. Let Bm and Bl be the blocks containing vx and vy respectively. If l m > 1 then f (v ) f (v )=(l m)n > n. Therefore, (v ,v ) E(I ). Now − 0 y − 0 x − x y 6∈ 0 we consider the case when l m 1. Consider the set of indifference graphs = − ≤ I Hj 1 j t that is generated by DET when given H as input. We know that { m | ≤ ≤ } m (v ,v ) E(H ) because H is an induced subgraph of G containing the vertices v and x y 6∈ m m x v . Since is a valid indifference graph representation of H , at least one of the graphs y I m in , say Hp , should not contain the edge (v ,v ). Recall that we denote by gp be the I m x y m p equal interval representation with interval length n for Hm that is constructed by DET. Since (v ,v ) E(Hp ), gp (v ) gp (v ) > n. Let i = m mod 3. Thus, m = 3r + i, x y 6∈ m | m x − m y | for some r 0. Now, since f is defined as the union of all the functions in the set ≥ i,p f gp : r = 0, 1, 2,... which contains gp , f (v )= gp (v ) and f (v )= gp (v ) { S} ∪ { 3r+i } m i,p x m x i,p y m y which implies that f (v ) f (v ) > n. Therefore, (v ,v ) E(I ). | i,p x − i,p y | x y 6∈ i,p

Thus, DETBAND generates a valid indiﬀerence graph representation of G using at Chapter 5. Cubicity and bandwidth 55

most 3t + 1 12(∆ + 1) ln(2b) + 1 indiﬀerence graphs. Since DET runs in polyno- ≤ ⌈ ⌉ mial time and there are only polynomial number of invocations of DET, the procedure DETBAND runs in polynomial time.

Tightness of the bound: Consider the case when G is a complete binary tree of height d = log n. Using the results shown in [14], we can see that cub(G) d = ≥ log 2d log n where c is a constant. Therefore, cub(G)=Ω( log n ). Since the bandwidth c1+log log n 1 log log n n of the complete binary tree on n vertices is Θ( log n ) as shown in [36], our O(∆ln b) bound on cubicity is tight up to a factor of O(log log n).

5.2 Cube representation in b + 1 dimensions

We shall now show that given a linear ordering of the vertices of G with width b, we can construct an indiﬀerence graph representation of G using b + 1 indiﬀerence graphs.

Theorem 5.4. If G is any graph with bandwidth b, then cub(G) b + 1. ≤ Proof: Let n denote V (G) and let = u , u ,...,u be a linear ordering of the | | A 0 1 n−1 vertices of G with width b. It is obvious that n > b. Since has width b, if (u , u ) A j k ∈ E(G), then j k b. For two vertices u , u V (G), we will abuse notation to say | − | ≤ j k ∈ that u < u if j < k and u > u if j > k. The relations and on V (G) are also j k j k ≤ ≥ defined similarly. We construct b + 1 indifference graphs I ,I ,...,I and H, such that G = I I 0 1 b−1 0 ∩ 1 ∩ I H. ···∩ b−1 ∩ Construction of indifference graph H: The vertex set of H is V (G) and let its edge set be denoted by E(H). Since H has to be a supergraph of G, we have to make sure that every edge in E(G) has to be present in E(H). b being the width of the linear ordering of vertices taken, a A vertex u is not adjacent in G to any vertex u when j k > b. Let the function j k | − | h : V (G) R be the equal interval representation for H with interval length b, i.e., for → u , u V (G), (u , u ) E(H) h(u ) h(u ) b. We construct h in such a way j k ∈ j k ∈ ⇔ | j − k | ≤ Chapter 5. Cubicity and bandwidth 56

that E(H)= (u , u ) j k < b (u , u ) j k = b and (u , u ) E(G) . h is { j k | | − | } ∪ { j k | | − | j k ∈ } deﬁned as: Let ǫ = 1/n2.

j, for j < b,

h(uj)=  h(u )+ b, for j b and (u , u ) E(G),  j−b j−b j  ≥ ∈  h(u )+ b + ǫ, for j b and (u , u ) E(G). j−b ≥ j−b j 6∈  

Note that for a vertex uj,

h(u ) h(u )+ b + ǫ j ≤ j−b h(u ) + 2b + 2ǫ h(u )+ j/b b + j/b ǫ ≤ j−2b ≤···≤ j mod b ⌊ ⌋ ⌊ ⌋ = j mod b + j/b b + j/b ǫ ⌊ ⌋ ⌊ ⌋ = j + j/b ǫ ⌊ ⌋ j + nǫ = j + 1/n ≤

Claim 4. H is a supergraph of G. Proof: First we observe that for any vertex u , j h(u ) j + 1/n. Now, consider j ≤ j ≤ an edge (u , u ) of G where j < k. Since the width of the input linear ordering is j k A b, we have k j b. Now we consider the following two cases. If k j b 1 − ≤ − ≤ − then h(u ) h(u ) k + 1/n j b 1 + 1/n < b. Since h(u ) h(u ) b, it k − j ≤ − ≤ − | k − j | ≤ follows that (u , u ) E(H). If k j = b then from the deﬁnition of h, it follows that j k ∈ − h(u )= h(u )+ b = h(u )+ b. Thus h(u ) h(u )= b implying that (u , u ) E(H). k k−b j k − j j k ∈ Every edge in G is therefore present in H, or in other words, H is a supergraph of G.

Construction of I , for 0 i b 1 : i ≤ ≤ − The vertex set of the indiﬀerence graph Ii is V (G) and let E(Ii) denote the edge set of I . I is constructed as follows. Let v ,v ,...,v be a subsequence of of i i 0 1 k−1 A k vertices such that v0 = ui, v1 = ui+b, v2 = ui+2b,...,vj = ui+jb and so on where Chapter 5. Cubicity and bandwidth 57

k = n−i . We deﬁne v as a dummy vertex with the property that u V (G),u

fi(u) = 1, if u < ui

If u be a vertex such that u u : ≥ i

fi(u) = t, if u = vt = t + 2, if v

Claim 5. I for 0 i b 1 is a supergraph of G. i ≤ ≤ − Proof: Consider the indiﬀerence graph Ii. Let (x, y) be any edge in E(G). We assume without loss of generality that x < y. Case x < u = v : Then y < u = v . Thus, f (x)=1 and 0 f (y) 3. Therefore, i 0 i+b 1 i ≤ i ≤ f (x) f (y) 2 which implies that (x, y) E(I ) (since f is an equal interval repre- | i − i | ≤ ∈ i i sentation with interval length 2 for Ii). Case x = v for some t k 1: Then y v , therefore f (x) = t and f (y) = t + 1 t ≤ − ≤ t+1 i i (if y = v ) or t + 2 (if y

It remains to show that G = I I H. To do this, it suﬃces to show that 0 ∩···∩ b−1 ∩ any (x, y) / E(G) is not present in at least one of the indiﬀerence graphs I ,...,I ,H. ∈ 0 b−1 Let x = uj and y = uk and we will assume without loss of generality that j < k (i.e. Chapter 5. Cubicity and bandwidth 58

x < y). Consider the case k j b. In this case, we claim that (x, y) / E(H). This is − ≥ ∈ because of the following. If k j = b then clearly h(x) h(y) = h(u ) h(u ) = b + ǫ − − k − j and thus (x, y) / E(H) (recall that h is an equal interval representation with interval ∈ length b). Now, if k j b +1 then h(u ) h(u ) k j 1/n (b + 1) 1/n > b − ≥ k − j ≥ − − ≥ − (since h(u ) k and h(u ) j + 1/n). Thus (u , u ) / E(H). Now the remaining case k ≥ j ≤ j k ∈ is k j < b. Consider the graph I where l = j mod b. Let t = j/b and let v = u , − l ⌊ ⌋ r l+rb for r = 0, 1, 2,.... Then v = u . Since k j < b, u < v . Thus we have f (u ) = t t j − k t+1 l j and f (u )= t + 3. Thus, f (u ) f (u ) > 2 and hence (u , u ) / E(I ) as required. l k | l j − l k | j k ∈ l Thus I0,...,Ib−1,H is a valid indiﬀerence graph representation of G using b + 1 indiﬀerence graphs which establishes that cub(G) b + 1. ≤

Tightness of the bound: Though the bound of cub(G) bw(G) + 1 might seem far ≤ from being tight for many graphs such as complete graphs, there are several graphs for which the bound becomes almost tight. For example, the bandwidth and cubicity of paths are both equal to 1 and for cycles, the bandwidth and cubicity are both equal to 2—our bound is thus tight but for an additive constant of 1. A Roberts’ graph is the graph obtained by removing a perfect matching from a complete graph. It can be seen from the results given in [51] that the cubicity of a Roberts’ graph on n vertices is n/2. The bandwidth of the Roberts’ graph can be seen to be n 2 upon observation. Thus − our bound is tight upto a factor of 2 for Roberts’ graphs.

The algorithm:

Our algorithm to compute the cube representation of G in b+1 dimensions given a linear ordering of the vertices of G with width b constructs the indiﬀerence supergraphs of G, namely, I0,...,Ib−1,H using the constructive procedure used in the proof of Theorem 5.4. It is easy to verify that this algorithm runs in O(b n) time where b is the width of · the input linear arrangement and n is the number of vertices in G. Chapter 5. Cubicity and bandwidth 59

5.3 Cubicity of special graph classes

Theorem 5.4 can be used to derive upper bounds for the cubicity of several special classes of graphs such as circular-arc graphs, cocomparability graphs and AT-free graphs. We ﬁnd upper bounds for the bandwidth of these graph classes in terms of the maximum degree and consequently obtain upper bounds on the cubicity. Bandwidth of circular- arc graphs have been studied in [42, 38], that of AT-free graphs in [39] and that of cocomparability graphs in [41]. The following lemmas can also be proved using certain properties given in [42, 39, 41].

5.3.1 Circular-arc graphs

Deﬁnition 5.5. Circular-arc graphs are the intersection graphs of intervals (or “arcs”) on a circle.

Figure 5.1 shows a circular-arc graph and its representation as the intersection of arcs on a circle.

v5 v3 v4

v5 v4

v7 v2 v1

v2 v6 v7 v3

Figure 5.1: A circular-arc graph: the graph on the right is the intersection graph of the circular-arcs on the left

Lemma 5.6. If G is a circular-arc graph, bw(G) 2∆, where ∆ is the maximum degree ≤ Chapter 5. Cubicity and bandwidth 60

of G. Proof: Let an arc on a circle corresponding to a vertex u be denoted by [h(u),t(u)] where h(u)(called the head of the arc) is the starting point of the arc when the circle is traversed in the clockwise order and t(u) (called the tail of the arc) is the ending point of the arc when traversed in the clockwise order. We assume without loss of generality that the end-points of all the arcs are distinct and that no arc covers the whole circle. If any of these cases occur, the end-points of the arcs can be shifted slightly so that our assumption holds true. Choose a vertex v V (G). Start from h(v ) and traverse the circle in the clockwise 1 ∈ 1 order. We order the vertices of the graph (other than v1) as v2,...,vn in the order in which the heads of their corresponding arcs are encountered during this traversal. Now, we deﬁne an ordering f : V (G) 1,...,n of the vertices of G as follows: → { } f(v ) = 2j, if 1 j n/2 . j ≤ ≤ ⌊ ⌋ f(v )=2(n j) + 1, if n/2 < j n. j − ⌊ ⌋ ≤ We now prove that the width of this ordering is at most 2∆.

We claim that if h(vj) and h(vk) are two consecutive heads encountered during a clockwise traversal of the circle, f(v ) f(v ) 2. To see this, we will consider the | j − k | ≤ diﬀerent cases that can occur: Case 1. When 1 j < j +1 = k n/2 . Here, f(v ) = 2j and f(v ) = 2(j + 1). ≤ ≤ ⌊ ⌋ j k Therefore, f(v ) f(v ) = 2. | j − k | Case 2. When n/2

Case 4. When j = n and k = 1. We then have f(vj) = 1 and f(vk) = 2. Therefore, f(v ) f(v ) = 1. | j − k | Chapter 5. Cubicity and bandwidth 61

Now, consider any edge (v ,v ) E(G). Assume without loss of generality that j k ∈ h(vj) occurs ﬁrst when we traverse the circle in clockwise direction starting from h(v1).

Now, if we traverse the arc corresponding to vj from h(vj) to t(vj), we will encounter at most ∆ 1 heads h(u ),h(u ),...,h(u ) before we reach h(v ) since v can be − 1 2 ∆−1 k j connected to at most ∆ vertices in G. We already know that f(v ) f(u ) 2 and | j − 1 | ≤ f(u ) f(u ) 2, for 1 i ∆ 2. Also, f(u f(v ) 2. It follows that | i − i+1 | ≤ ≤ ≤ − | ∆−1 − k | ≤ f(v ) f(v ) 2∆. Thus f is an ordering of the vertices of G with width at most 2∆ | j − k | ≤ and therefore we have bw(G) 2∆. ≤

Corollary 5.7. If G is a circular-arc graph with maximum degree ∆, then cub(G) ≤ 2∆ + 1. Proof: Follows from Theorem 5.4 and Lemma 5.6.

5.3.2 Cocomparability graphs

Deﬁnition 5.8. Comparability graphs are graphs that have a transitive orientation.

That is, the edges of such a graph G can be oriented to obtain a directed graph G~ so that if there is a directed path from u to v in G~ then the directed edge (u, v) is present in G~ . As an exercise, it is instructive to verify that Cn (a cycle on n vertices) is a comparability graph if and only if n is even.

Deﬁnition 5.9. Cocomparability graphs are graphs whose complements are comparability graphs.

Lemma 5.10. If G is a cocomparability graph, then bw(G) 2∆ 1, where ∆ is the ≤ − maximum degree of G. Proof: Let V (G) = n. Since G is a comparability graph, there exists a partial order | | ≺ in G on the node set V (G) such that (u, v) E(G) if and only if u v or v u. This ∈ ≺ ≺ Chapter 5. Cubicity and bandwidth 62

partial order gives a direction to the edges in E(G). We can run a topological sort on this partial order to produce a linear ordering of the vertices, say, f : V (G) 1,...,n . → { } The topological sort ensures that if u v, then f(u) < f(v). Now, let (u, v) E(G) and ≺ ∈ let w be a vertex such that f(u) < f(w) < f(v). We will show that w is adjacent to either u or v in G. Suppose not. Then (u, w), (w,v) E(G) and therefore u w and w v. ∈ ≺ ≺ Now, by transitivity of , this implies that u v, which means that (u, v) E(G)—a ≺ ≺ ∈ contradiction. Therefore, any vertex w such that f(u) < f(w) < f(v) in the ordering f is adjacent to either u or v. Since the maximum degree of G is ∆, there can be at most 2∆ 2 vertices between with f( ) value between f(u) and f(v). Thus, the width of the − · ordering given by f is at most 2∆ 1 and therefore, bw(G) 2∆ 1. − ≤ −

Corollary 5.11. If G is a cocomparability graph with maximum degree ∆, then cub(G) 2∆. ≤ Proof: Follows from Theorem 5.4 and Lemma 5.10.

5.3.3 AT-free graphs

AT-free graphs were deﬁned in Section 1.2 (see Deﬁnition 1.6).A caterpillar is a tree such that a path (called the spine) is obtained by removing all its leaves (see Figure 5.2). In the proof of Theorem 3.16 of [39], Kloks et al. show that every connected AT-free

Figure 5.2: An example of a caterpillar: the white vertices indicate the spine Chapter 5. Cubicity and bandwidth 63

graph G has a spanning caterpillar subgraph T , such that adjacent nodes in G are at a distance at most four in T . Moreover, for any edge (u, v) E(G) such that u and v are ∈ at distance exactly four in T , both u and v are leaves of T . Let p1,...,pk be the nodes along the spine of G.

Lemma 5.12. If G is an AT-free graph, bw(G) 3∆ 2, where ∆ is the maximum ≤ − degree of G. Proof: Let L denote the set of leaves of T adjacent to p . Clearly, L ∆ and i i | i| ≤ L L = for i = j. For any set S of vertices, let S denote an arbitrary ordering of i ∩ j ∅ 6 h i the vertices in set S. Let u denote ordering with just one vertex u in it. If α = u ,...,u h i 1 s and β = v ,...,v are two orderings of vertices in G, then let α β denote the ordering 1 t ⋄ u ,...,u ,v ,...,v . Let = L p L p L p be a linear ordering 1 s 1 t A h 1i⋄h 1i⋄h 2i⋄h 2i⋄···⋄h ki⋄h ki of the vertices of G. One can use the property of T stated in the previous paragraph to easily show that is a linear ordering of the vertices of G with width at most 3∆ 2. A − Therefore, bw(G) 3∆ 2. ≤ −

Corollary 5.13. If G is an AT-free graph with maximum degree ∆, then cub(G) ≤ 3∆ 1. − Proof: Follows from Theorem 5.4 and Lemma 5.12.

5.4 A summary of results

The upper bounds for cubicity we have presented so far are summarized in the following table: Chapter 5. Cubicity and bandwidth 64

Graph class Upper bound for cubicity

Any graph 12(∆ + 1) ln(2b) + 1 ⌈ ⌉ Any graph 4(∆ + 1) ln n ⌈ ⌉ Any graph b + 1 AT-free graphs 3∆ 1 − Interval graphs ∆ + 1 Circular arc graphs 2∆ + 1 Cocomparability graphs 2∆

Note that AT-free graphs include well-known graph classes like interval graphs, permutation graphs and trapezoidal graphs. It is well known that interval graphs have bandwidth at most ∆ and hence the upper bound for interval graphs. However, this bound is far from being tight as we see in the next chapter. Chapter 6

Cubicity of interval graphs

Interval graphs are a very well studied class of graphs not just because of their well- defined structure but also because of their usefulness in a wide variety of applications ranging from DNA analysis to process scheduling. Since interval graphs have boxicity at most 1, the result in [15] gives us an upper bound of log n for the cubicity of any ⌈ 2 ⌉ interval graph on n vertices. Theorem 5.4 gives us an upper bound of ∆ + 1 for any interval graph with maximum degree ∆. We now show that the special structure of these graphs can be exploited to show that a much tigher upper bound exists for their cubicity when compared to either of these bounds. We prove constructively that for any interval graph I on n vertices with maximum degree ∆, cub(I) log ∆ + 4. More specifically, an indifference graph representation ≤ ⌈ 2 ⌉ of I is constructed using log ∆ + 4 indifference graphs. ⌈ 2 ⌉

6.1 A few results that we need

Two lemmas that we need for the proof follow.

Lemma 6.1. For an interval graph I, there exists an interval representation such that the intervals assigned to no two vertices have the same left end-point. Proof: Consider an interval representation of I and let l(u) and r(u) denote the left and right end-points of the interval assigned to vertex u by this interval representation. We

65 Chapter 6. Cubicity of interval graphs 66

construct a new interval representation of I by mapping each vertex u to a new interval [l′(u), r(u)] such that l′(u) = l′(v) for any two vertices u and v. We deﬁne l′(u) as follows. 6 Let P = x R u V such that l(u)= x or r(u)= x . Let ǫ = 1 min x y . { ∈ | ∃ ∈ } n+1 x,y∈P {| − |} Let v ,...,v be an ordering of the vertices of I such that if i < j then l(v ) l(v ) 1 n i ≤ j (resolving ties arbitrarily). For a vertex v , deﬁne l′(v )= l(v ) (n + 1 i)ǫ. It is easy i i i − − to verify that for any two vertices v and v , l′(v ) = l′(v ). In the following we show that i j i 6 j ′ mapping each vertex vi to [l (vi), r(vi)] yields a valid interval representation of I. Clearly for any vertex v , l′(v ) r(v ). Consider any two vertices v and v and let i < j. i i ≤ i i j Recalling the ordering, we know that l(v ) l(v ) since i < j. It is easy to see that i ≤ j (v ,v ) is an edge in I if and only if r(v ) l(v ). It is easy to verify that under the new i j i ≥ j mapping, l′(v ) l′(v ). Moreover, r(v ) l′(v ) if and only if r(v ) l(v ). It follows i ≤ j i ≥ j i ≥ j ′ ′ that for vi and vj, their corresponding new intervals [l (vi), r(vi)] and [l (vj), r(vj)] have a non-empty intersection if and only if their corresponding original intervals [l(vi), r(vi)] and [l(vj), r(vj)] have a non-empty intersection.

A construction to obtain the indiﬀerence graph representation of any interval graph on n vertices using log n indiﬀerence graphs was given in [15]. We state the lemma and ⌈ 2 ⌉ describe in brief the construction involved because we use the result and the underlying construction to prove the main result in this chapter.

Lemma 6.2 (Chandran and K. A. Mathew [15]). If G is an interval graph on n vertices, cub(G) log n . ≤ ⌈ 2 ⌉ The construction in [15]: The proof of this lemma gives an indifference graph representation of G using k = log n indifference graphs I ,...,I . The construction of ⌈ 2 ⌉ 1 k each I , 1 i k involves partitioning the vertex set into two sets A and B . For i ≤ ≤ i i each Ii, its equal interval representation with interval length n (recall Definition 1.16) is specified by a function h : V [0, 2n] in such a way that h (u) [0, n] for all u A i → i ∈ ∈ i and h (u) [n + 1, 2n] for all u B . We can assume the existence of f : V [0, 2], a i ∈ ∈ i i → unit interval representation of I . (As noted in Section 1.3, the function f : V [0, 2] i i → such that fi(u)= hi(u)/n is a unit interval representation of Ii.) Chapter 6. Cubicity of interval graphs 67

6.2 The proof

Let I(V,E) be an interval graph on n vertices with maximum degree ∆. We assume n 2 and ∆ 2 as the remaining cases are trivial. Consider an interval representation ≥ ≥ of I which assigns the interval [l(u), r(u)] to each vertex u V (G). By Lemma 6.1 we can ∈ assume that this given interval representation of I is such that l(u) = l(v) for all u, v 6 ∈ V (I) where u = v. 6 Below we state a useful property of interval graphs.

Lemma 6.3. Let (u, v) E(I) such that l(u) < l(v) and let S = w l(u) < l(w) < ∈ { | l(v) . Then d(u) S + 1. } ≥ | | Proof: This is so because we have r(u) l(v) since (u, v) E(I) and therefore ≥ ∈ l(w) [l(u), r(u)] for each w S. Thus u is adjacent to all the vertices in S (in addition ∈ ∈ to v).

6.2.1 Grouping the vertices

Let v1,v2,...,vn be an ordering of the vertices of I such that if i < j, then l(vi) < l(vj). Now, we group the vertices into disjoint blocks B , B ,...,B where k = n/(2∆) . Each 1 2 k ⌈ ⌉ block except possibly the last consists of 2∆ vertices. That is, B = v ...,v i { 2(i−1)∆+1 2i∆} for 1 i k 1 and B = v ,...,v . Let the block number of a vertex u, ≤ ≤ − k { 2(k−1)∆+1 n} denoted as b(u), be deﬁned as follows. For 1 i k and for all u B , deﬁne b(u)= i. ≤ ≤ ∈ i For 1 i k 1, let the block B be partitioned into two parts X and Y where ≤ ≤ − i i i

X = u B N(u) B = and Y = B X i { ∈ i | ∩ i+1 ∅} i i − i

Let X = B and Y = . k k k ∅ Lemma 6.4. Let (u, v) E(I) such that l(u) < l(v). If b(u) = b(v) then u Y and ∈ 6 ∈ b(u) v X . ∈ b(u)+1 Proof: Suppose b(u) = b(v). First we show that b(v) = b(u) + 1. In other words, we 6 Chapter 6. Cubicity of interval graphs 68

show that v B . Clearly b(v) > b(u) since l(v) > l(u) and b(u) = b(v). Assume for ∈ b(u)+1 6 the sake of contradiction that b(v) > b(u) + 1. It follows that for all w B , l(u) < ∈ b(u)+1 l(w) < l(v). By Lemma 6.3, it implies that d(u) B + 1. Since b(v) > b(u) + 1, ≥ | b(u)+1| we have b(u) + 1 k 1 and hence B = 2∆. It follows that d(u) 2∆ + 1 which ≤ − | b(u)+1| ≥ contradicts the fact the ∆ is the maximum degree of I. Now it is easy to see from the deﬁnition of Y that u Y since v B and b(u) ∈ b(u) ∈ b(u)+1 v N(u). ∈ It remains to be shown that v X . If b(u)+1 = k, then we are done. Let ∈ b(u)+1 ′ b(u)+1= t < k. Let Bt be partitioned as Zt and Zt where

Z = v ,...,v and Z′ = B Z t { 2(t−1)∆+1 2(t−1)∆+∆} t t − t

Recalling that B = 2∆, we have Z = Z′ = ∆. First we note that v Z . This | t| | t| | t| ∈ t is because, if v Z′ then for all w Z , l(u) < l(w) < l(v). This would imply that ∈ t ∈ t d(u) ∆+1 by Lemma 6.3, which is a contradiction. Now we show that N(v) B = . ≥ ∩ t+1 ∅ This is because, if say p N(v) B then clearly for all w Z′, l(v) < l(w) < l(p). ∈ ∩ t+1 ∈ t This would imply by Lemma 6.3 that d(v) ∆ + 1, which is a contradiction. Thus we ≥ have v X . ∈ t

Corollary 6.5. Let (u, v) E(I) such that l(u) < l(v). If u X then b(u)= b(v). ∈ ∈ b(u)

Lemma 6.6. Let u and v be two vertices of I such that b(u)= b(v)= b. If (u, v) / E(I) ∈ then either u X or v X . ∈ b ∈ b Proof: If b = k, then the theorem is trivially true as Xk = Bk. Therefore, we consider the case when b < k. Assume for the sake of contradiction that u, v Y . From the ∈ b deﬁnition of set Y , we have N(u) B = and N(v) B = . Let u′ N(u) B b ∩ b+1 6 ∅ ∩ b+1 6 ∅ ∈ ∩ b+1 and let v′ N(v) B . Let w V such that l(w) = min l(x) . It follows that ∈ ∩ b+1 ∈ x∈Bb+1 { } l(u) l(w) l(u′) and l(v) l(w) l(v′). Since (u, u′), (v,v′) E(I), we also have ≤ ≤ ≤ ≤ ∈ l(u′) r(u) and l(v′) r(v). It follows that l(w) [l(u), r(u)] and l(w) [l(v), r(v)]. In ≤ ≤ ∈ ∈ Chapter 6. Cubicity of interval graphs 69

other words, the intervals corresponding to u and v have non-empty intersection, which contradicts the fact that (u, v) / E(I). Thus, either u X or v X . ∈ ∈ b ∈ b

6.2.2 Constructing the indiﬀerence graph representation

Let t = log (2∆) = log (∆) + 1. We now construct t + 3 indiﬀerence graphs ⌈ 2 ⌉ ⌈ 2 ⌉ ′ ′ ′ H1,H2,...,Ht,H0,H1 and H2 such that

t I = H′ H′ H′ H . 0 ∩ 1 ∩ 2 ∩ i i=1 \ Let us define these indifference graphs by giving their unit interval representations. Let j, for 1 j k, denote the subgraph induced by X on I. Clearly, j is an I ≤ ≤ j I interval graph on at most 2∆ vertices. Also, j is not empty as X is not empty for I j any j. This is because it follows from Lemma 6.6 that Yj induces a clique in I. Now, for j < k, if X = , then Y = 2∆. Thus Y induces a clique of size 2∆ in I j ∅ | j| j which is an obvious contradiction to the fact that ∆ is the maximum degree in I. If j = k, Xj = Bj by definition and hence not empty. Now, from Lemma 6.2, we have cub( j) log (2∆) = t. This means that j can be represented as the intersection of I ≤ ⌈ 2 ⌉ I t indifference graphs, say, j, j,..., j. From the construction described in Section 6.1, I1 I2 It it follows that there is a unit interval representation of j given by f j : X [0, 2]. Ii i j →

Construction of indiﬀerence graphs H1,...,Ht:

We deﬁne the unit interval representation of H , for 1 i t by the function g : V i ≤ ≤ i → [0, 2] as follows: Let u be a vertex of I. Let b = b(u). That is u belongs to the block Bb. For 1 i t: ≤ ≤ g (u)= f b(u) if u X and g (u) = 1 if u Y i i ∈ b i ∈ b Recalling the deﬁnition of f b(), it follows that g (u) [0, 2] for any vertex u. i i ∈ Lemma 6.7. For i 1,...,t , H is a supergraph of I. ∈ { } i Chapter 6. Cubicity of interval graphs 70

Proof: Consider H for i 1,...,t . Let (u, v) E(I) such that l(u) < l(v). We show i ∈ { } ∈ that (u, v) E(H ). If u Y , then by definition of g , g (u) = 1. Since g (v) lies in ∈ i ∈ b(u) i i i [0, 2], we have g (u) g (v) 1 and therefore (u, v) E(H ). By similar reasoning, | i − i | ≤ ∈ i it follows that if v Y then (u, v) E(H ). The only remaining case is that both ∈ b(v) ∈ i u X and v X . From corollary 6.5, we have b(u) = b(v) = b. That is, u X ∈ b(u) ∈ b(v) ∈ b and v X . Hence u, v V ( b). Since (u, v) E(I) and hence (u, v) E( b), the edge ∈ b ∈ I ∈ ∈ I (u, v) is present in all the indifference graphs b, b,..., b. In particular, (u, v) E( b). I1 I2 It ∈ Ii Thus, we have, f b(u) f b(v) 1. Now, from the definition of g , we have g (u)= f b(u) | i − i | ≤ i i i and g (v)= f b(v). Thus, g (u) g (v) 1 implying that (u, v) H . i i | i − i | ≤ ∈ i

Lemma 6.8. Let (u, v) E(I) such that u, v X where l = b(u) = b(v). Then 6∈ ∈ l (u, v) E( t H ). 6∈ i=1 i l Proof: ClearlyT u and v are vertices in the induced subgraph induced by Xl on I. Since I (u, v) / E(I), we have (u, v) / E( l). Recalling that t l is an indifference graph ∈ ∈ I i=1 Ii representation of l, it follows that there exists a j 1,...,tT such that (u, v) / E( l). I ∈ { } ∈ Ij In other words, the unit intervals corresponding to u and v in l, given by f l(u) and Ij j f l(v), are such that f l(u) f l(v) > 1. We now show that (u, v) / H , implying that j | j − j | ∈ j (u, v) / E( t H ). To see this, first recall that u, v X . Hence, in the unit interval ∈ i=1 i ∈ l representationT of Hj, the intervals corresponding to u and v, given by gj(u) and gj(v), are such that g (u)= f l(u) and g (v)= f l(v), by definition. It follows that g (u) g (v) > 1 j j j j | j − j | implying that (u, v) / H . ∈ j

′ ′ Construction of H0 and H1:

For i 0, 1 , let g′ : V R specify the unit interval representation of H′. ∈ { } i → i Consider a vertex u V . Let b = b(u) be the block to which u belongs. Deﬁne ∈ S(u) as S(u) = X if u X and S(u) = Y otherwise. (Either u X or u Y .) Let b ∈ b b ∈ b ∈ b p(u)= x S(u) l(x) l(u) . Let ǫ = 1/n. |{ ∈ | ≤ }| ′ Deﬁne g0 as follows: Chapter 6. Cubicity of interval graphs 71

g′ (u)= b + ǫ p(u) if u Y 0 · ∈ b g′ (u)= b 1 if u X and Y N(u)= 0 − ∈ b b ∩ ∅ g′ (u)= b 1+ ǫ p(n ) if u X and Y N(u) = , where 0 − · u ∈ b b ∩ 6 ∅

p(nu) = maxx∈Yb∩N(u) p(x) ′ { } Similarly, deﬁne g1 as: g′ (u)= b + ǫ p(u) if u X 1 · ∈ b g′ (u)= b + ǫ p(n ) if u Y , where p(n ) = max p(x) . 1 · u ∈ b u x∈Xb+1∩N(u){ } (Note that if u Y , then X N(u) = ). ∈ b b+1 ∩ 6 ∅ ′ ′ Lemma 6.9. H0 and H1 are supergraphs of I. Proof: Let (u, v) E(I) such that l(u) < l(v). We show that (u, v) E(H′ ) and ∈ ∈ 0 (u, v) E(H′ ) by proving that g′ (u) g′ (v) 1 and g′ (u) g′ (v) 1. Note that ∈ 1 | 0 − 0 | ≤ | 1 − 1 | ≤ b(u) b(v) since l(u) < l(v). ≤ Case b(u)= b(v)= b where either u, v X or u, v Y : ∈ b ∈ b It is straightforward to verify from the deﬁnition of g′ that g′ (u) g′ (v) 1 (by 0 | 0 − 0 | ≤ observing that 0 < ǫ p(w) 1 for any w V ). · ≤ ∈ Case b(u)= b(v)= b where either u X and v Y or u Y and v X : ∈ b ∈ b ∈ b ∈ b If u X and v Y then by noting that g′ (u)= b(u) 1+ǫ p(n ) and p(v) p(n ), it ∈ b ∈ b 0 − · u ≤ u follows that g′ (u) g′ (v) 1. The symmetric case u Y and v X follows similarly. | 0 − 0 | ≤ ∈ b ∈ b Case b(u) = b(v) : 6 In this case, by Lemma 6.4 we have b(v) = b(u)+1 with u Y and v X . ∈ b(u) ∈ b(u)+1 Now, by noting that b(u) g′ (u) b(u)+1 and b(u) g′ (v) b(u) + 1, it follows that ≤ 0 ≤ ≤ 0 ≤ g′ (u) g′ (v) 1. | 0 − 0 | ≤ Using similar arguments, it is straightforward to show that g′ (u) g′ (v) 1 for the | 1 − 1 | ≤ ′ ′ above three cases. Thus, H0 and H1 are supergraphs of I.

Lemma 6.10. Let u and v be two vertices of I such that b(u) = b(v) and v Y . If ∈ b(v) (u, v) E(I) then (u, v) E(H′ ). 6∈ 6∈ 0 Chapter 6. Cubicity of interval graphs 72

Proof: Let b(u) = b(v) = b. Since v Y , we have b < k. From Lemma 6.6 we have ∈ b u X . In the following, we show that g′ (u) g′ (v) > 1, which would imply that ∈ b | 0 − 0 | (u, v) / E(H′ ). Since v Y , we have g′ (v)= b + ǫ p(v). Since u X , if Y N(u)= ∈ 0 ∈ b 0 · ∈ b b ∩ ∅ then it is easy to verify that g′ (u) g′ (v) > 1, and thus (u, v) / E(H′ ). | 0 − 0 | ∈ 0 Consider the remaining case Y N(u) = . Recall that n Y N(u) such that b ∩ 6 ∅ u ∈ b ∩ p(n ) = max p(x) . We ﬁrst show that p(v) >p(n ) as follows. Recalling that u x∈Yb∩N(u){ } u u X and v Y , we have N(u) B = and N(v) B = . It easily follows ∈ b ∈ b ∩ b+1 ∅ ∩ b+1 6 ∅ that r(v) > r(u) because for any w N(v) B , r(v) l(w) and r(u) < l(w) (since ∈ ∩ b+1 ≥ N(u) B = ). Since r(v) > r(u) and (u, v) / E(I), it follows that l(u) < l(v). Since ∩ b+1 ∅ ∈ n N(u) we also have r(u) l(n ). If l(v) l(n ), we would obtain that l(u) < l(v) u ∈ ≥ u ≤ u ≤ l(n ) r(u), which would imply that (u, v) E(I), which is a contradiction. Hence u ≤ ∈ it follows that l(v) > l(n ). Since v, n Y , it follows from the deﬁnition of p( ) that u u ∈ b · p(v) >p(n ). Recalling that g′ (u)= b 1+ ǫ p(n ) and g′ (v)= b + ǫ p(v), it follows u 0 − · u 0 · that g′ (v) g′ (u) = 1+ ǫ (p(v) p(m)) > 1. Therefore, (u, v) E(H′ ). | 0 − 0 | | · − | 6∈ 0

Lemma 6.11. Let u and v are two vertices of I such that b(v)= b(u) + 1, u Y and ∈ b(u) v X . If (u, v) E(I) then (u, v) E(H′ ). ∈ b(v) 6∈ 6∈ 1 Proof: We have l(u) < l(v) since b(v) = b(u)+1. Let b(u) = b and b(v) = b + 1. We show that (u, v) / E(H′ ) by showing that g′ (u) g′ (v) > 1. If X N(u)= , from ∈ 1 | 1 − 1 | b+1 ∩ ∅ the deﬁnition of g′ , it is straightforward to verify that g′ (u) g′ (v) > 1, implying that 1 | 1 − 1 | (u, v) H′ . 6∈ 1 Consider the remaining case X N(u) = . Recall that n X N(u) such b+1 ∩ 6 ∅ u ∈ b+1 ∩ that p(n ) = max p(x) . Note that l(u) < l(n ) since b(n ) = b(u) + 1. u x∈N(u)∩Xb+1 { } u u Since n N(u) and l(u) < l(n ), we have l(u) < l(n ) r(u). If l(v) < l(n ) then, u ∈ u u ≤ u recalling that l(u) < l(v), it follows that l(u) < l(v) < l(n ) r(u), implying that u ≤ (u, v) E(I), which is a contradiction. Thus we have l(v) > l(n ). Since v, n Y , it ∈ u u ∈ b follows from the deﬁnition of p( ) that p(v) > p(n ). Finally we have g′ (v) g′ (u) = · u | 1 − 1 | 1+ ǫ (p(v) p(m)) > 1. Therefore, (u, v) E(H′ ). | · − | 6∈ 1 Chapter 6. Cubicity of interval graphs 73

′ Construction of the indiﬀerence graph H2:

Let g′ : V R denote the unit interval representation of H′ . 2 → 2 Deﬁne g′ as follows: Let u V . 2 ∈ g′ (u) = 2b(u) 1 if u X 2 − ∈ b(u) g′ (u) = 2b(u) if u Y 2 ∈ b(u) ′ Lemma 6.12. H2 is a supergraph of I. Proof: Consider an edge (u, v) E(I) such that l(u) < l(v). We show that g′ (u) ∈ | 2 − g′ (v) 1, which implies that (u, v) E(H′ ). Let b(u) = b. If b(u) = b(v) = b 2 | ≤ ∈ 2 then clearly g′ (u) g′ (v) 1. If b(u) = b(v) then it follows from Lemma 6.4 that | 2 − 2 | ≤ 6 b(v)= b(u)+1= b +1 and u Y and v X . Thus, g′ (u) = 2b and g′ (v) = 2b + 1, ∈ b ∈ b+1 2 2 implying that g′ (u) g′ (v) 1. Therefore, (u, v) E(H′ ). | 2 − 2 | ≤ ∈ 2

Lemma 6.13. Let u and v be two vertices such that b(v) > b(u) and (u, v) E(I). 6∈ If b(v) > b(u) + 1, then (u, v) E(H′ ). Also, if b(v) = b(u) + 1 and if u Y or 6∈ 2 6∈ b(u) v X , then (u, v) E(H′ ). 6∈ b(v) 6∈ 2 Proof: We show that for any such u and v, g′ (v) g′ (u) > 1, which implies that | 2 − 2 | (u, v) / E(H′ ). ∈ 2 ′ Consider the ﬁrst case b(v) > b(u) + 1. It is clear from the deﬁnition of g2 that g′ (v) g′ (u) > 1. The remaining case is b(v) = b(u) + 1. If u Y then g′ (u) = | 2 − 2 | 6∈ b(u) 2 2b(u) 1. But g′ (v) 2b(v) 1 = 2b(u)+1. It follows that g′ (v) g′ (u) > 1. If v X − 2 ≥ − | 2 − 2 | 6∈ b(v) then g′ (v) = 2b(v) = 2b(u)+2. But g′ (u) 2b(u) and therefore g′ (v) g′ (u) > 1. 2 2 ≤ | 2 − 2 | Thus (u, v) / E(H′ ). ∈ 2

6.2.3 The theorem

Theorem 6.14. Given an interval graph I with maximum degree ∆, cub(I) ≤ log ∆ + 4. ⌈ 2 ⌉ Proof: Recall that t = log (2∆) = log (∆) +1. We show that I = t H 2 H′, ⌈ 2 ⌉ ⌈ 2 ⌉ i=1 i ∩ i=0 i which by Lemma 1.19 implies our theorem. From Lemmas 6.7, 6.9 andT 6.12, weT know Chapter 6. Cubicity of interval graphs 74

′ ′ ′ that each of H1,...,Ht,H0,H1,H2 is a supergraph of I. It remains to show that if (u, v) / E(I) then (u, v) is not present in at least one of ∈ ′ ′ ′ the indiﬀerence graphs H1,...,Ht,H0,H1,H2.

Case b(u)= b(v) and u, v X : ∈ b(u) In this case, from Lemma 6.8, we have, (u, v) E( t H ). 6∈ i=1 i Case b(u)= b(v), and either v Y or u Y : T ∈ b(v) ∈ b(u) If v Y then from Lemma 6.10, we have (u, v) E(H′ ). Clearly, the symmetric ∈ b(v) 6∈ 0 case u Y also follows from Lemma 6.10. ∈ b(u) Now for the remaining case of b(u) = b(v) we assume without loss of generality that 6 b(v) > b(u).

Case b(v)= b(u) + 1: If u Y and v X then from Lemma 6.11 we have (u, v) E(H′ ). If u / Y ∈ b(u) ∈ b(v) 6∈ 1 ∈ b(u) or v / X then from Lemma 6.13 we have (u, v) E(H′ ). ∈ b(v) 6∈ 2 Case b(v) > b(u) + 1: In this case, from Lemma 6.13 we have (u, v) E(H′ ). 6∈ 2

Tight example:

Consider the star graph S = K . It was shown in [51] that cub(S) = log n . The 1,n ⌈ 2 ⌉ maximum degree of S being n, we have cub(S) = log ∆(S) . This shows that our ⌈ 2 ⌉ upper bound of log ∆ + 4 is tight up to the additive constant of 4. ⌈ 2 ⌉

6.3 Remarks

It follows from the results of Booth and Lueker [12] that interval graphs can be rec- ognized in polynomial time and that an interval representation can be constructed in polynomial time for interval graphs. Thus, given an interval graph, its cube representation in log ∆ + 4 dimensions can also be computed in polynomial time. It should be ⌈ 2 ⌉ Chapter 6. Cubicity of interval graphs 75

noted that our result does not improve the upper bound of log n on cub(G) since the ⌈ 2 ⌉ box(G) maximum degree of each interval graph in an interval graph representation of G could be as large as n 1. −

Chapter 7

Planar graphs

The boxicity of planar graphs was shown to be at most 3 by Thomassen [57]. A better bound holds for outerplanar graphs, a subclass of planar graphs. Scheinerman [54] showed that the boxicity of outerplanar graphs is at most 2. But this bound does not hold for the class of series-parallel graphs, a slightly bigger subclass of planar graphs than the outerplanar graphs. Bohra et al. [8] showed that there exists series-parallel graphs with boxicity 3. In this chapter, we present an independent proof for the fact that outerplanar graphs have boxicity at most 2.

7.1 Preliminaries

The plane drawing of a graph refers to a drawing of a graph on the plane such that no two edges cross each other. Planar graphs are exactly those graphs that have plane drawings. The plane drawing of a graph splits the plane into regions (contiguous sets of points enclosed between the edges of the graph) called “faces”. A face is said to be bounded if it is possible to draw a large enough circle that contains the whole face. Otherwise, it is unbounded. In every plane drawing, there is exactly one face that is unbounded, called the “outermost face”. Pagenumber of a graph: Consider drawing a graph as follows. Arrange the vertices

77 Chapter 7. Planar graphs 78

of the graph in some order along the spine of a book and draw each edge on a page of the book in such a way that no two edges on the same page cross each other. Such a drawing is called a book drawing of the graph (see Figure 7.1). The minimum number of pages required for a book drawing of a graph is called the pagenumber or book thickness of the graph.

Figure 7.1: A book drawing of K5 using 3 pages

Book thickness of a graph was deﬁned in 1979 by Bernhart and Kainen [6]. It was shown by Yannakakis [64] that planar graphs have pagenumber at most 4. The following lemma is fairly straightforward.

Lemma 7.1. Let G be a graph with n vertices. G has pagenumber 1 if and only if there is an arrangement v ,...,v of the vertices of G such that i,j,k,l i

7.2 Outerplanar graphs

Deﬁnition 7.2. Outerplanar graphs are planar graphs which have a plane drawing such that all the vertices lie on the boundary of the outermost face.

The following lemma is from [6].

Lemma 7.3 (Bernhart and Kainen [6]). Outerplanar graphs have pagenumber at most 1. We shall now show that every outerplanar graph has an interval graph representation using two interval graphs and therefore has boxicity at most 2.

Theorem 7.4. Outerplanar graphs have boxicity at most 2. Proof: Let G be an outerplanar graph on n vertices. G therefore has pagenumber 1 by Lemma 7.3. Consider a book drawing of G with one page in which the ordering of vertices along the spine is given by v ,...,v . For any vertex v , deﬁne N (v ) = j n j > 1 n i r i { | ≥ i and (v ,v ) E(G) . Similarly, deﬁne N (v ) = j 1 j

i, if Nr(vi)= right(vi)= ∅  max N (v ) , otherwise  { r i } and  i, if Nl(vi)= left(vi)= ∅  min N (v ) , otherwise.  { l i }

We shall construct two interval graphs I1 and I2 by deﬁning their interval representations f1 and f2 respectively as follows:

i, 1 i n, f (v ) = [i,right(v )] ∀ ≤ ≤ 1 i i

i, 1 i n, f (v ) = [left(v ),i] ∀ ≤ ≤ 2 i i

Claim 1. I1 and I2 are supergraphs of G. Proof: Let (v ,v ) E(G). Assume without loss of generality that i < j. Clearly, i j ∈ Chapter 7. Planar graphs 80

j N (v ) and hence right(v ) j. Thus we have l(f (v )) < l(f (v )) r(f (v )) ∈ r i i ≥ 1 i 1 j ≤ 1 i and therefore f (v ) f (v ) = implying that (v ,v ) E(I ). Similarly, i N (v ). 1 i ∩ 1 j 6 ∅ i j ∈ 1 ∈ l j Thus, left(v ) i. We therefore get l(f (v )) r(f (v )) < r(f (v )) which means that j ≤ 2 j ≤ 2 i 2 j (v ,v ) E(I ). i j ∈ 2

Claim 2. If (v ,v ) E(G), then (v ,v ) E(I ) or (v ,v ) E(I ). i j 6∈ i j 6∈ 1 i j 6∈ 2 Proof: Let (v ,v ) E(G). Assume without loss of generality that j < k. Suppose j k 6∈ (v ,v ) E(I ). Therefore, f (v ) f (v ) = implying that l(f (v )) < l(f (v )) j k ∈ 1 1 j ∩ 1 k 6 ∅ 1 j 1 k ≤ r(f (v )). Thus, k right(v ). This means that there exists some v with l > k such 1 j ≤ j l that (v ,v ) E(G). Now, we claim that (v ,v ) E(I ). Suppose for the sake of j l ∈ j k 6∈ 2 contradiction that (v ,v ) E(I ). Then we have f (v ) f (v ) = implying that j k ∈ 2 2 j ∩ 2 k 6 ∅ l(f (v )) r(f (v )) < r(f (v )). Thus, left(v ) j which means that there exists 2 k ≤ 2 j 2 k k ≤ some v with i < j such that (v ,v ) E(G). Now, we have i

It follows from Claims 1 and 2 that G = I I . Hence the theorem. 1 ∩ 2

7.3 Discussion

The pagenumber of a graph is a property that one would like to relate with the boxicity. But some facts should be noted: complete graphs are a class of graphs with boxicity smaller than the pagenumber and outerplanar graphs are a class of graphs with pagenumber smaller than the boxicity. Again, K2,2,2 can be seen to be a graph with boxicity 3 and pagenumber 2 and at the same time planar graphs need boxicity at most 3 but there are planar graphs that need pagenumber 4 [63]. Still, the problem is interesting as both boxicity and pagenumber are bounded for planar graphs; a possible hint to some geometric connection between the two parameters. Chapter 8

Boxicity of Halin graphs

8.1 A short introduction

For a graph G =(V,E), we write G = T C if E(G)= E(T ) E(C) where T is a tree on ∪ ∪ the vertex set V (G) and C is a simple cycle on the leaves of T . Such a graph G is called a Halin graph if G has a planar embedding and T has no vertices of degree 2. Figure 8.1 shows an example of a Halin graph. The notion of Halin graphs were ﬁrst used by

Figure 8.1: A Halin graph: the tree edges are in bold

Halin [34] in his study of minimally 3-connected graphs. Bondy and Lovasz [11] proved that these graphs are almost pancyclic—they contain a cycle of each length between 3 and n with the possible exception of one length, which must be even. Bondy [10] has

81 Chapter 8. Boxicity of Halin graphs 82

also shown that Halin graphs are 1-Hamiltonian—i.e, they are Hamiltonian and if any one vertex or edge from the graph is removed, the resulting graph is also Hamiltonian. Lovasz and Plummer [44] show that every Halin graph with an even number of vertices is minimal bicritical (a graph is bicritical if the removal of any two vertices from the graph will result in a graph with a perfect matching). Halin graphs are also interesting because some problems that are NP-complete for general graphs have been shown to be polynomial-time solvable for Halin graphs. Examples are the travelling salesman problem [22] and the problem of ﬁnding a dominating cycle with at most l vertices [56]. It has been shown in [58] that every Halin graph is a 2-interval graph—i.e., the intersection graph of sets, each of which is the union of at most 2 intervals. We show in this chapter that the boxicity of a Halin graph (not isomorphic to K4) is equal to 2 which means that every Halin graph is the intersection graph of axis-parallel rectangles on the plane (in other words, Cartesian products of two intervals) as well. In fact, we show a stronger result—we show that our result holds for any graph G = T C that has ∪ a planar embedding, even if there are vertices of degree 2 in T . Since box(G) = 1 when

G is isomorphic to K4, we show our result for graphs not isomorphic to K4. We know that planar graphs need boxicity at most 3 [57]. It was proved in the last chapter that outerplanar graphs, a subclass of planar graphs, need boxicity at most 2. We show here that Halin graphs, another subclass of planar graphs, need boxicity at most 2. Quest and Wegner [50] have characterized the graphs with boxicity at most 2 using the adjacency matrix and the “induced C–V matrices” of a graph. But as far as we can see, there is no straightforward way to use this characterization on Halin graphs to obtain the result presented here.

8.2 The proof

Let G = T C where C is a simple cycle connecting the leaves of a tree T such that ∪ G is planar. Our strategy will be to construct two interval graphs G1 and G2 such that G = G G thus proving that boxicity of G is at most 2. It can be easily seen that a 1 ∩ 2 Chapter 8. Boxicity of Halin graphs 83

cycle has boxicity 2 unless it is a triangle (in which case it has boxicity 1) and a wheel being just a universal vertex added to a cycle, has boxicity 2 unless it is a K4 (in which case it has boxicity 1). Therefore, we will assume that G is not a wheel. For the sake of ease of presentation, a vertex will be called a “leaf” or “leaf vertex” if it is a leaf of the tree T . Given H V (G), we denote by G the subgraph induced by the vertices of H ⊆ H in G. Since T is a tree, there is unique path between any two vertices u and v in T . We denote this path by uT v.

8.2.1 Finding u′

Let S = V (G) V (C) denote the set of internal vertices of the tree T . We claim that − there is a vertex u′ S such that N(u′) S = 1 and N(u′) V (C) 1. If there is no ∈ | ∩ | | ∩ | ≥ such vertex, then GS, the induced subgraph of G on S, has no vertices of degree 1 which is not possible since GS is a tree (GS has more than one vertex since G is not a wheel). Now, u′ has at least one leaf of T as its neighbour since if it did not, then its degree in T is 1 implying that u′ is a leaf of T —a contradiction since we have assumed that u′ S. ∈

8.2.2 Fixing the root of T

Designate the internal vertex of T adjacent to u′, say r, to be the root of T . Given two vertices u and v, u is said to be an ancestor of v if u lies in the path rTv and u is said to be a descendant of v if v is an ancestor of u. Note that every vertex is an ancestor and a descendant of itself. Let D(u) for any vertex u V (G) be deﬁned as the set of all ∈ leaves of T that are descendants of u. It can be easily seen that if u is a descendant of v, then D(u) D(v). ⊆

8.2.3 Ordering the vertices of C

Let V (C) = k and let C be p p ...p p . Note that D(u′) cannot contain all the | | 0 1 k−1 0 leaves since that would mean that D(u′)= D(r), implying that u′ is the only neighbour of r in T . Then the degree of r in T would be 1, a contradiction since r is an internal Chapter 8. Boxicity of Halin graphs 84

vertex in T and not a leaf. Therefore, we can always find a leaf p D(u′) such that i ∈ p D(u′) (recall that u′ has at least one leaf of T as its neighbour and there- (i−1) mod k 6∈ fore, D(u′) is not empty). We define l = p , for 0 j k 1. This implies j (i+j) mod k ≤ ≤ − that l D(u′) since l = p . For u V (C), we define c(u)= i when u = l . k−1 6∈ k−1 (i−1) mod k ∈ i

For the convenience of the reader, we summarize the construction as of now:

We chose a vertex u′ such that its neighbourhood contains exactly one internal • vertex and at least one leaf of T .

We chose the only internal vertex in the neighbourhood of u′ to be the root r of T • and deﬁned the natural tree-order on T with r as the root. We also deﬁned D(u) to be the set of all leaves that are descendants (in our tree-order) of the vertex u.

We deﬁned a linear ordering l ,...,l of the vertices in V (C) (the leaves of T ) • 0 k−1 where l D(u′) and l D(u′). 0 ∈ k−1 6∈

Lemma 8.1. For any vertex u V (G), the vertices in D(u) will occur in consecutive ∈ places in the ordering l ,...,l of the vertices in C. In other words, if u V (G) and 0 k−1 ∈ x, y, z V (C) such that c(x) < c(z) < c(y) then it is not possible that x, y D(u) and ∈ ∈ z D(u). 6∈ Proof: If u is a leaf of T , then the lemma is true because D(u) = 1. Let us assume | | that this is not the case. Consider any planar embedding of G. The cycle C divides the plane into a bounded region and an unbounded region. We claim that all the internal vertices of T will lie in one of these regions. Suppose there are two internal vertices of T such that they lie in diﬀerent regions of C. Then, the path between them in T will have to pass the boundary of C. But the path cannot pass through a leaf of T and because the drawing is planar, no edge of the path can cross the boundary of C. We thus have a contradiction. Therefore, C forms the boundary of a face in any planar drawing of G. Now, consider a planar embedding of G such that C forms the boundary of the unbounded face (i.e., all the internal vertices of T lie in the bounded region of C). Chapter 8. Boxicity of Halin graphs 85

Suppose x, y D(u) and z D(u) such that c(x) < c(z) < c(y) (recall that c(l ) = i). ∈ 6∈ i Let B = xCyT uT x. It can be easily veriﬁed that B has exactly two regions—one bounded and the other unbounded. We say that a vertex is “inside” B if it lies in the bounded region bounded by B and say that it is “outside” B if it lies in the unbounded region whose boundary is B. We say that a vertex “lies on” B if it is in B. Observation 1. Because of the planar embedding of G that we have chosen, it can be seen that any leaf vertex will have to either lie on xCy or outside B. Observation 2. r does not lie on B. We can assume that r = u since that would contradict our assumption that z D(u). 6 6∈ Also, r cannot lie on yT u or uT x since it contradicts our assumption that x and y are descendants of u and it cannot lie on xCy since it is not a leaf. Therefore, r does not lie on B. Observation 3. u′ is not inside B.

′ ′ If u is inside B, then l0 cannot be outside B since u is adjacent to l0. From Obser- vation 1, l is in xCy which implies that x = l (since 0 c(x) c(v), for any vertex 0 0 ≤ ≤ v xCy, as c(x) < c(y)) and u′, being the only internal vertex in N(l ), should lie on ∈ 0 uT x. This contradicts our assumption that u′ is inside B. Observation 4. r is outside B. Now suppose r is inside B. Then, u′ cannot be outside B since r is adjacent to u′ and it cannot be inside B due to Observation 3. Therefore, u′ lies on B. If u = u′, 6 then the fact that r is the only internal vertex adjacent to u′ implies that r will have to lie on B, which contradicts Observation 2. Therefore, u = u′. Now, it can be seen that because of our choice of u′ and r, D(u′)= N(u′) r . This means that uT x and − { } uT y are the edges u′x and u′y respectively and therefore, any path from r (inside B)

′ to a vertex outside B will have to go through u . Now, consider the leaf lk−1. By our construction, l D(u′). Therefore, y = l and l does not lie on xCy and hence k−1 6∈ 6 k−1 k−1 lies outside B (from Observation 1). The path from r to lk−1 will have to go through u′ as we have noted before—but this implies that l D(u′) which is a contradiction. k−1 ∈ Therefore, r is outside B since we know from Observation 2 that r does not lie on B. Chapter 8. Boxicity of Halin graphs 86

Because of Observation 4, the path zT r must contain a vertex v in B because of our planarity assumption. But if v = u, then x and y cannot both be descendants of u 6 since either rT x or rT y will not contain u. If v = u, then rT z contains u and therefore, z D(u), again a contradiction. ∈ This proves our claim that for any vertex u V (G), the vertices in D(u) have to ∈ occur consecutively in the ordering l0, l1,...,lk−1.

8.2.4 Construction of the interval graphs G1 and G2

We deﬁne f1 and f2 to be mappings of the vertex set V (G) to closed intervals on the real line. Let G1 and G2 denote the interval graphs deﬁned by f1 and f2 respectively. For a vertex u V (G), let d(u) denote the number of ancestors of u other than itself ∈ (or “depth” of u in T ). Let h denote the maximum depth of a vertex in T . Recall that k = V (C) and S denotes the set of internal vertices of T . | |

Deﬁnition of f1: For u V (G), ∈ f1(l0) = [0, k]. f (u) = [c(u) 1/2,c(u) + 1/2], if u V (C) and u = l . 1 − ∈ 6 0 f (u) = [min c(v) , max c(v) ], if u S. 1 v∈D(u){ } v∈D(u){ } ∈

Deﬁnition of f2: For u V (G), ∈ ′ ′ f2(u ) = [d(u ),h + 2] = [1,h + 2]. f (u) = [d(u),d(u) + 1], if u S and u = u′. 2 ∈ 6 f2(l0) = [h + 2,h + 2].

f2(l1) = [d(l1),h + 2].

f2(lk−1) = [d(lk−1),h + 2]. f (u) = [d(u),h + 1], if u V (C) and u is not l , l or l . 2 ∈ 0 1 k−1 Chapter 8. Boxicity of Halin graphs 87

Lemma 8.2. G1 is a super graph of G. Proof: Consider an edge (u, v) E(G). Clearly, (u, v) E(T ) or (u, v) E(C). ∈ ∈ ∈ 1. (u, v) E(T ). ∈ In this case, either u is an ancestor of v or vice versa as T is a tree. Let us assume without loss of generality that u is the ancestor of v. Therefore, D(v) D(u). ⊆ There are two possibilities now:

(a) u and v are both internal vertices of T . Since D(v) D(u), we have min c(x) min c(x) ⊆ x∈D(u){ } ≤ x∈D(v){ } ≤ max c(x) max c(x) . Therefore, f (u) f (v) = , which x∈D(v){ } ≤ x∈D(u){ } 1 ∩ 1 6 ∅ implies that (u, v) E(G ). ∈ 1 (b) u is an internal vertex of T and v is a leaf vertex of T . Since v D(u), min c(x) c(v) max c(x) . Thus, both ∈ x∈D(u){ } ≤ ≤ x∈D(u){ } f (u) and f (v) contain the point c(v) and therefore, (u, v) E(G ) (Note 1 1 ∈ 1 that c(l ) = 0 and thus c(l ) f (l )). 0 0 ∈ 1 0 2. (u, v) E(C). ∈ Without loss of generality, we can assume that u = li, for some i, and v = l . For 1 i k 2, f (u) and f (v) contain the point i + 1/2. If u = l (i+1) mod k ≤ ≤ − 1 1 0 or v = l , then it is clear that (u, v) E(G ), since f (l ) contains f (u), u V (G). 0 ∈ 1 1 0 1 ∀ ∈

Therefore, G1 is a supergraph of G.

Lemma 8.3. G2 is a supergraph of G. Proof: Consider an edge (u, v) E(G). We have the following three cases now. ∈

1. u or v is l0. ′ By our choice of l0, it is adjacent only to l1, lk−1 and u in G. Since f2(l0), ′ f2(l1), f2(lk−1) and f2(u ) contain the point h + 2, all the edges incident on l0 in G

are also present in G2. Chapter 8. Boxicity of Halin graphs 88

2. (u, v) E(T ), u = l and v = l . ∈ 6 0 6 0 Let us assume without loss of generality that u is the parent of v. It is easily seen that d(v)= d(u) + 1. Since u = l and v = l , the point d(u) + 1 is contained 6 0 6 0 in both f (u) and f (v) (Recall that d(u) h, u V (G)). 2 2 ≤ ∀ ∈ 3. (u, v) E(C), u = l and v = l . ∈ 6 0 6 0 Since u and v are leaf vertices, f2(u) and f2(v) both contain the point h + 1 and therefore (u, v) E(G ). ∈ 2

This shows that G2 is a supergraph of G.

Lemma 8.4. G = G G . 1 ∩ 2 Proof: Since Lemmas 8.2 and 8.3 have established that G1 and G2 are supergraphs of G, it is suﬃcient to show that, for any pair of vertices u, v V (G), (u, v) E(G) implies ∈ 6∈ (u, v) E(G )or(u, v) E(G ). Consider such a pair of vertices. There are three cases 6∈ 1 6∈ 2 to be considered.

1. One of u or v is l0. Let us assume without loss of generality that u = l . (u, v) E(G) now implies 0 6∈ that v V (G) l , l , u′ since l is only adjacent to l , l and u′ in G. It can ∈ − { 1 k−1 } 0 1 k−1 ′ be easily veriﬁed that only f2(u ), f2(l1) and f2(lk−1) have a non-empty intersection with f (l ). Therefore, (u, v) E(G ). 2 0 6∈ 2 2. u = l , v = l and one of u and v is the ancestor of the other. 6 0 6 0 Let us assume without loss of generality that u is the ancestor of v. This implies that d(v) d(u) + 2 since (u, v) E(G). We know that u = u′ since ≥ 6∈ 6 all the descendants of u′ are its neighbours by our choice of u′ and the root r. Now, since u = u′, the right end-point of f (u) is d(u) + 1 and for all possible 6 2 choices of v (excluding l ), the left end-point of f (v) is d(v) d(u)+2. Therefore, 0 2 ≥ f (u) f (v)= by the deﬁnition of f . Thus, in this case, (u, v) E(G ). 2 ∩ 2 ∅ 2 6∈ 2 3. u = l , v = l and neither one of u and v is an ancestor of the other. 6 0 6 0 One of the following three subcases hold. Chapter 8. Boxicity of Halin graphs 89

(a) u and v are both leaves of T .

Let u = li and v = lj. Assume without loss of generality that i < j. Since neither of u or v is l , we have 1 i < j k 1. Also, j>i + 1 as 0 ≤ ≤ − (u, v) E(G). Therefore, f (l ) f (l )= , from the definition of f . Thus, 6∈ 1 i ∩ 1 j ∅ 1 we have (u, v) E(G ). 6∈ 1 (b) u and v are both internal vertices of T . Since u rTv and v rT u, we have D(u) D(v)= (To see this, suppose 6∈ 6∈ ∩ ∅ there is a vertex z D(u) D(v). Then both u and v would lie on rT z, ∈ ∩ implying that either u rTv or v rT u). Now, from Lemma 8.1, we have ∈ ∈ max c(x) < min c(x) or max c(x) < min c(x) . x∈D(u){ } x∈D(v){ } x∈D(v){ } x∈D(u){ } By the definition of f , it can be seen that f (u) f (v) = , implying that 1 1 ∩ 1 ∅ (u, v) E(G ). 6∈ 1 (c) One of u and v is a leaf of T and the other is an internal vertex of T . Let us assume that u is an internal vertex and v is a leaf of T . Since we are considering the case when neither of u and v is an ancestor of the other and neither is l , we have v D(u) and v = l . From Lemma 8.1, we know 0 6∈ 6 0 that either c(v) < min c(x) or c(v) > max c(x) . Therefore, x∈D(u){ } x∈D(u){ } by definition of f and because v = l , f (u) f (v) = and thus we have 1 6 0 1 ∩ 1 ∅ (u, v) E(G ). 6∈ 1 Since we have considered all possible cases when (u, v) E(G) and have shown that 6∈ in each case, (u, v) is not present either in E(G ) or in E(G ), it follows that G = G G . 1 2 1 ∩ 2

Now, to complete the proof, we show that if G is not isomorphic to K , then box(G) 4 ≥ 2. Suppose G is not isomorphic to K4. We will show that G is not an interval graph. By deﬁnition of G, V (C) 3. If V (C) > 3, then C is an induced cycle with more than | | ≥ | | 3 vertices which means that G cannot be an interval graph and therefore box(G) 2. ≥ If V (C) = 3, then C is a triangle. Now, all the leaves in V (C) cannot be adjacent to | | the same internal vertex of T . To see this, look at GS, the subgraph induced by S in Chapter 8. Boxicity of Halin graphs 90

G (recall that S = V (G) V (C), or the set of internal vertices of T ). Since G is not − isomorphic to K4, GS is a tree with more than one vertex. Therefore, there are at least two vertices of degree 1 in GS. But if all the vertices in V (C) are adjacent only to one vertex of S in G, there should be at least one vertex in S with degree 1 in G—which is a contradiction since all vertices of S, being internal vertices of T , have degree more than 1 in G. Therefore, we can ﬁnd two leaves, say x and y, of T such that they are adjacent to diﬀerent internal vertices in T . Let u and v denote the internal vertices of T adjacent to x and y respectively. Now, xuT vyx forms an induced cycle of length greater than or equal to 4 (note that x and y are adjacent since V (C) = 3). Therefore, G cannot be | | an interval graph. Thus, we have box(G) 2. ≥

8.3 Results

From the discussion in the last section, we have the main result of this chapter in the form of the following theorem.

Theorem 8.5. If G = T C, where T is a tree and C is a simple cycle of the leaves ∪ of T such that G is planar, then box(G) = 2 if G is not isomorphic to K4.

Corollary 8.6. Every Halin graph has boxicity equal to 2 unless it is isomorphic to

K4, in which case it has boxicity equal to 1. Chapter 9

Conclusion

9.1 Improvements

Some of the results presented in this thesis have been since improved. A short survey of these and other related results follows. In Chapter 2, we showed that for any graph G with maximum degree ∆, box(G) ≤ 2∆2. This upper bound has been improved to box(G) ∆2 + 2 for any graph G by ≤ Esperet [28]. That paper improves upon the basic idea of colouring the graph by using a more sophisticated colouring scheme and a modiﬁed interval graph representation to achieve the better bound. The upper bound on the cubicity of interval graphs presented in Chapter 6 has also been improved. In a recent unpublished work, Adiga and Chandran [1] show that for any interval graph G, cub(G) log ψ(G) + 2 where ψ(G) is deﬁned as the largest ≤ ⌈ 2 ⌉ integer m such that K1,m, the star graph with m arms, is an induced subgraph of G. Note that ψ(G) ∆ where ∆ is the maximum degree of G and therefore, this is a much ≤ tigher bound when compared to our upper bound of log ∆ + 4. ⌈ 2 ⌉

91 Chapter 9. Conclusion 92

9.2 Open problems

9.2.1 Boxicity and maximum degree

We conjectured in Chapter 2 that the boxicity of any graph with maximum degree is O(∆). The conjecture is still open and any progress towards proving or disproving the conjecture would be exciting.

9.2.2 The boxicity of hypercubes

Chandran and Sivadasan show in [17] that the cubicity of the d-dimensional hyper cube

d d Hd is Θ( log d ). This automatically shows that box(Hd) is O( log d ). We do not know of any tighter upper bound for the boxicity of the d-dimensional hypercube. The problem of whether a tighter upper bound exists seems to be an interesting one.

9.2.3 Cubicity of planar graphs

We know that any planar graph has boxicity at most 3 [57]. Now, what about the cubicity? The result from [15] gives us an upper bound of 3 log n for the cubicity of ⌈ 2 ⌉ any planar graph on n vertices but is it the best possible bound? In terms of n only, we cannot hope to achieve a bound better than log n as the star graph is a planar graph ⌈ 2 ⌉ with cubicity log n . But a better bound might be possible in terms of other graph ⌈ 2 ⌉ parameters.

9.2.4 Algorithms for computing the boxicity

In many practical applications, the graphs that arise have some special structure that can be utilized for speeding up the computations on them. Although it is NP-hard to compute the boxicity of general graphs, we could restrict ourselves to special graph classes and see if the problem becomes polynomial-time solvable. Chapter 9. Conclusion 93

9.2.5 Hard problems on bounded boxicity graphs

We have seen in Section 1.5.1 that the max-clique problem is polynomial-time solvable if the box representation of the input graph in a bounded number of dimensions is available. It was also mentioned that better approximation algorithms could be constructed for some problems with the assumption that a box or cube representation of the input graph in a bounded number of dimensions is available. What other hard problems become easier to solve given a low dimensional box or cube representation is worth studying.

9.2.6 Fixed-parameter tractable algorithms

Fixed-parameter tractable or FPT algorithms typically solve hard problems in O(f(k)nc) time where n is the input size, c is a constant and k is an input parameter that depends on the problem instance. f can be any function that is deﬁned solely in terms of k. The idea is that if the input parameter for all the problem instances that we need to solve is small, then the algorithm performs well for all the required problem instances. A number of FPT algorithms use the treewidth of the input graph as a parameter. Chandran and Sivadasan [16] showed that box(G) tw(G) + 2. As noted before, the ≤ max-clique problem is polynomial-time solvable if the box representation of the input graph in a bounded number of dimensions is available. It might therefore be possible that FPT algorithms could be constructed with the boxicity of the graph as a parameter.

9.3 Endnote

The author is aware that this thesis is more of a collection of results on boxicity and cubicity than a comprehensive guide to these topics. A large body of literature is available for the interested reader who wishes to pursue the study of geometric intersection graphs and a number of references are listed in the bibliography that follows. Though primarily deﬁned in terms of the intersection of geometric objects, the combinatorial nature of boxicity and cubicity are evinced by their relationship with parameters Chapter 9. Conclusion 94

such as the partial order dimension. Being natural generalizations of the widely studied class of interval graphs, and possessing a neat geometric intersection model, the class of intersection graphs of boxes and cubes oﬀers an exciting direction of research. Generalizations of other geometric intersection models might also be attempted.

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