## NSF/CBMS Regional Research Conference in Mathematical Sciences on Geometric Graph Theory

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Geometric Graph Theory

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## May 28 - June 1, 2002 Gateway Center

University of North Texas, Denton

## Geometric Graph Theory

The first genuine monograph on graph theory (König, 1936) had the following subtitle: Combinatorial Topology of Systems of Segments [K]. Although graph theory and topology stem from the same root, the connection between them has somewhat faded away in the past few decades. In the most prolific new areas of graph theory including Ramsey theory, extremal graph theory and random graphs, graphs are regarded as abstract binary relations rather than systems of segments. It is quite remarkable that traditional graph theory is often incapable of providing satisfactory answers for the most natural questions concerning the drawings of graphs. Geometric Graph Theory is a relatively new discipline at the borderline of discrete mathematics and computer science, which was developed to deal with such questions. During the past 10 years Geometric Graph Theory has played a significant role in settling and solving several fundamental problems of combinatorial and computational geometry. We believe that a better understanding of this field will be crucial in answering many other important questions in discrete mathematics and in the theory of computing.

A geometric graph G = (V(G),E(G)) is a graph drawn in the plane by line segments, i.e., V(G) is a set of points in the plane, no three of which are collinear, and E(G) is a set of segments whose endpoints belong to V(G). A topological graph is defined similarly, except that its edges can be arbitrary Jordan curves connecting two elements of V(G) and not passing through a third vertex. Two edges of a topological (or a geometric) graph are said to cross if they have a point in common, different from their endpoints.

The study of geometric graphs was initiated by Erdos [AE], Avital and Hanani [AH], Perles [AP] and Kupitz [Ku] more than 30 years ago. In the early eighties, topological and geometric graphs were intensively studied by the theoretical computer science community, due to their applications in VLSI design. In 1989, during the Special Year on Discrete and Computational Geometry at DIMACS, Perles gave a one-semester course on the subject. János Pach, at the time a Visiting Professor at DIMACS, attended these lectures, became interested in the topics and soon became one of the leading experts in this area.

One of the main objectives in geometric graph theory is to characterize all possible crossing patterns determined by the edges of a given geometric graph [G1,T2] or, more generally, by a collection of continuous arcs (strings'') in the plane. We are very far from having satisfactory solutions to these problems, but the existing results are truly fascinating! They are intimately related to fundamental results in topology [HLS,RS], in linear algebra, in the theory of block designs [BF], and in extremal set theory [EH]. They have many applications to problems in combinatorial and computational geometry, including the k-set problem [ELSS], proximity problems [S], and the problem of bounding the number of incidences between elementary geometric objects such as points and lines, curves, surfaces [PSo,ST1,CEG].

There are two famous old unsolved problems dealing with crossings: Turán's Brick-Factory Problem [T1] and Conway's Thrackle Conjecture [W].

The Brick Factory problem (which came to Turán's mind at a factory yard, where, as forced labour during World War II, he moved wagons filled with bricks from kilns to storage places) can be formulated as follows. Given an abstract graph G, find a drawing of G (i.e., a topological graph isomorphic to G) for which the number of crossings is minimum. This number is called the crossing number of G and is denoted by cr(G). Specifically, Turán's (still unsolved) original problem was to determine cr(Kn,m), for the complete bipartite graph Kn,m with n and m vertices in its classes, for every n, m 3. According to an assertion of Zarankiewicz, which was down-graded from a theorem to a conjecture [G], we have

This conjecture is still open. In fact, we do not know even the limits

for Kn,n and for the complete graph Kn.

A thrackle is a topological graph so that any two edges have exactly one point in common: they either meet at a common endpoint or they cross. Conway's conjecture states that the number of edges of a thrackle cannot exceed the number of vertices. At first glance this may appear to be just an innocent puzzle from recreational mathematics. However, this is likely to represent the tip of a huge iceberg, and a solution to the problem will probably take us several steps closer to understanding the structure of crossings, which may be viewed as a planar analogue of Knot Theory. Lovász, Pach, and Szegedy [LPS], who verified the conjecture up to a multiplicative factor of 2, found that parity arguments play an important role in this subject, just like in the case of the Brick Factory Problem. There is a remarkable theorem of Hanani [H] and Tutte [T2] behind this phenomenon: if any two edges of a topological graph, not incident to the same vertex, cross an even number of times, then the underlying abstract'' graph is planar, i.e., it can be drawn with no crossings.

Crossing numbers have also been studied in theoretical computer science due to their applications in VLSI design. In the early eighties, Leighton [L,BL] and others proved that the chip area required for the realization of an electrical network is closely related to the crossing number of the underlying graph. This discovery gave an impetus to research in the subject. It was soon realized that main difficulty in this problem is that a graph has so many essentially different drawings that the computation of the crossing number, even for a graph of only 15 vertices, appears to be a hopelessly difficult task. Garey and Johnson proved that the determination of the crossing number is NP-complete [GJ]. Therefore, approximating algorithms, heuristic methods, and general estimates play important roles in the subject.

One of the most applicable estimates for crossing numbers is an inequality discovered by Ajtai, Chvátal, Newborn, and Szemerédi [AC], and independently by Leighton [L]: The crossing number of any graph with n vertices and e > 4n edges is at least a constant times . The best known value of the constant was found by Pach and Tóth [PT]. A fundamental result of Pach, Spencer, and Tóth [PST] settled an old conjecture of Erdos and Guy [EG] by showing essentially that there exists an asymptotically best constant'' for which this inequality holds as n tends to infinity, no matter how fast e goes to infinity at the same time. They also improved the inequality for special classes of graphs having some monotone properties. Their estimates are asymptotically tight and generalize to graphs drawn on other surfaces.

It was a surprising discovery by Székely [S] that this lower bound easily implies a number of famous theorems of Beck, Chung, Szemerédi, Trotter and others [Be1,C,ST1,ST2,CST,SST,CEG] concerning some hard questions of Erdos in Combinatorial Geometry. In particular, it yields that the number of unit distances determined by n points in the plane is O(n4/3). Shortly after, Dey [D] proved that the same approach can be used for establishing an O(n4/3) bound for the number of different ways of splitting a set of n points in the plane into two equal parts by a line. This was a sensational improvement of an old result of Lovász [Lo], with many important applications in Computational Geometry. We emphasize that this problem or, more generally, the problem of bounding the number of k-element subsets ("k-sets") of an n-element point set in d-space, which can be separated from the remaining points by a hyperplane, arises in the analysis of virtually every geometric algorithm dealing with arrangements of points and hyperplanes. Another exciting recent development in this field is that, using the same technique, Solymosi and Tóth [SoT] have improved an important theorem of Chung, Szemerédi and Trotter [CST], and Székely [S]. They showed that any set of n points in the plane determines at least (n6/7) distinct distances. One may hope that this bound can be improved to about (n8/9) using the current methods. However, this would still be very far from the conjectured optimum, which is about .

There is another lower bound on the crossing number, derived by Leighton, which follows from a weighted version of the Lipton-Tarjan Separator Theorem [LT] for planar graphs. The bisection width of an abstract graph G, denoted by b(G), is the minimum number of edges whose removal partitions the vertex set into two parts such that there are no edges running between them and the larger set has at most twice as many vertices as the smaller. According to a more general form of Leighton's inequality, proved by Pach, Shahrokhi and Szegedy [PSS], the bisection width of any graph G with n vertices satisfies

where d1, d2, . . ., dn denote the degrees of the vertices. This result has many applications in computer graphics, computational geometry and graph drawing. For illustration, we mention a few examples.

I. Souvaine and Wenger studied the following problem arising in cartography [SW]. Given two disjoint rectangles P and Q in the plane, containing the points { p1, p2, . . ., pn } and { q1, q2, . . ., qn }, respectively, we would like to construct a piecewise linear isomorphism from P to Q, which maps pi to qi for every i. The objective is to minimize the complexity of such an isomorphism, i.e., the number of linear pieces it consists of. Souvaine and Wenger [SW] designed an algorithm for constructing an isomorphism with complexity O(n2). Pach, Shahrokhi and Szegedy [PSS] applied (*) to show that this bound cannot be improved.

II. Pach and Wenger [PW] proved that every planar graph with n vertices can be drawn in the plane so that the positions of the vertices are arbitrarily prescribed, and each edge is represented by a polygonal path consisting of at most 24n segments. The inequality (*) can be used to show that this result, too, is essentially tight.

III. Shahrokhi et al [SSS] applied (*) to analyze an efficient algorithm for constructing a geometric graph isomorphic to a given abstract graph G such that the number of crossings is O( log4 ncr(G)) and the coordinates of all vertices are integers bounded by O(n2). Even et al [EGS] have recently improved this analysis. In some sense, these results show that the simplest geometric and topological graphs isomorphic to G cannot be drastically different.

IV. M. Watanabe posed the following problem. Suppose we have a self-intersecting closed polygon P on the screen of our computer, whose vertices are p1, p2, . . ., pn in this order, and no three vertices are collinear. We are allowed to modify P so that in each step we can grab a vertex and move it to an arbitrary new position. Is it true that every polygon P can be untangled, i.e., turned into a noncrossing polygon, in at most steps, for some absolute constant < 1? Using (*), Pach and Tardos [PT] proved that the answer is in the negative.

V. A set of pairwise crossing edges in a geometric or topological graph is called a crossing set. Pach, Shahrokhi, and Szegedy [PSS] used (*) to prove that if a geometric graph with n vertices has no crossing set of size k, then its number of edges is at most 3n(10 log n)2k - 2. Valtr [V] improved this result by reducing the exponent of the logarithmic factor to 1. It is conjectured that for any k there exists a constant ck such that any geometric graph with no crossing set of size k has at most ckn edges. For k = 2 this follows from Euler's Polyhedral Formula, and for k = 3 it has been verified by Agarwal et al [AAP]. A closely related conjecture is that any complete geometric graph with n vertices has a crossing set of size (n). Aronov et al [AEG] proved the existence of a crossing set of size .

One of the most prolific and most applicable areas of (abstract) graph theory is Extremal Graph Theory [B], which was started by Turán's theorem on the maximum number of edges that a graph with n vertices can have without containing a complete subgraph with k vertices. It is interesting to note that (V) suggests that by asking analogous questions in a geometric setting, one may obtain exciting difficult problems, whose solutions require new techniques.

Two edges of a geometric graph are said to be disjoint if they do not share an endpoint or an interior point. A collection of pairwise disjoint edges is called a disjoint set. This notion is dual to that of a crossing set. In the spirit of Turán's theorem, we can ask what is the maximum number of edges that a geometric graph with n vertices can have without containing a disjoint set of size k? Hopf and Pannwitz [HP] proved that for k = 2 the answer is n. This implies that Conways's Thrackle Conjecture (mentioned above) is true for straight-line thrackles. It had been a longstanding open problem of Erdos and Perles whether the answer is at most ckn, for every k, before Pach and Törocsik settled this question in the affirmative. In their argument, ckn = O(k4) Tóth and Valtr [TV] and Tóth [T] improved this bound to
ckn = O(k3) and ckn = O(k2), respectively. It is possible that ckn is linear in k, but this may be difficult to prove. Results of this kind have interesting applications to the Map Labelling Problems [AKS] arising in Geographic Information Systems (GIS).

In a ground breaking series of papers, Pach and his co-authors have started to develop the Ramsey Theory of Geometric and Topological Graphs [KPT1,KPT2,KPT3,LMP,PSoT,PSo]. The prototype of the results in this area is that, no matter how we color the edges of a complete geometric graph with two colors, at least one of the color classes will always contain a noncrossing spanning tree. Results of this kind relate to deep problems for partially ordered sets [Tr].

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AAP
P.K. Agarwal, B. Aronov, J. Pach, R. Pollack, M. Sharir: Quasi planar graphs have linear number of edge, Combinatorica 17 (1997), 1-9.

AKS
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AC
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AE
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AP
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AEG
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AH
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BF
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BFP
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Be1
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Be2
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BL
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B
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C
F. R. Chung: The number of different distances determined by n points in the plane, J. Combin. Theory Ser. A 36 (1984), 342-354.

CST
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CEG
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D
T. K. Dey: Improved bounds for planar k-sets and related problems, Discrete and Computational Geometry 19 (1998), 373-382.

EG
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EH
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ELSS
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EGS
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FPP
H. de Fraysseix, J. Pach, and R. Pollack: How to draw a planar graph on a grid, Combinatorica 10 (1990), 41-51.

GJ
M. R. Garey and D. S. Johnson: Crossing number is NP-complete, SIAM Journal of Algebraic and Disccrete Methods 4 (1983), 312-316.

Gr
R. L. Graham: Problem, in: Combinatorics, Vol. II (A. Hajnal and V. T. Sós, eds.), North-Holland Publishing Company, Amsterdam, 1978, 1195.

G
R. K. Guy: The decline and fall of Zarankiewicz's theorem, in: Proof Techniques in Graph Theory, Academic Press, New York, 1969, 63-69.

G1
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H
A. Hanani (Ch. Chojnacki): Über wesentlich unplättbare Kurven im dreidimensionalen Raume, Fund. Math. 23 (1934), 135-142.

HLS
H. van der Holst, L. Lovász, and A. Schrijver: The Colin de Verdière graph parameter, in: Graph theory and combinatorial biology (Balatonlelle, 1996), Bolyai Soc. Math. Stud. 7, János Bolyai Math. Soc., Budapest, 1999, 29-85.

HP
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KPT1
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KPT2
G. Károlyi, J. Pach, G. Tóth, and P. Valtr: Ramsey-type results for geometric graphs. II, in: ACM Symposium on Computational Geometry (Nice, 1997), Discrete Comput. Geom. 20 (1998), 375-388.

KPT3
G. Károlyi, J. Pach, G. Tardos, and G. Tóth: An algorithm for finding many disjoint monochromatic edges in a complete 2-colored geometric graph, in: Intuitive Geometry (I. Bárány, K. Böröczky, eds.) Bolyai Society Mathematical Studies 6, Budapest, 1997, 367-372.

K
D. König: Theorie der endlichen und unendlichen Graphen, Leipzig, 1936. English translation: Theory of Finite and Infinite Graphs, Birkhäuser Verlag, Boston, 1990.

KM
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Ku
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LMP
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L
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LT
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Lo
L. Lovász: On the number of halving lines, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 14 (1971), 107-108.

LPS
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P1
J. Pach and P.K. Agarwal: Combinatorial Geometry, J. Wiley and Sons, New York, 1995.

P2
J. Pach: Geometric graph theory, in: Surveys in Combinatorics, 1999 (J. D. Lamb and D. A. Preece, eds.), London Mathematical Society Lecture Note Series 267, Cambridge University Press, Cambridge, 1999, 167-200.

P3
J. Pach: A problem of Ulam on planar graphs, European J. Combin. 2 (1981), 357-361.

PS
J.Pach and M. Sharir: On the number of incidences between points and curves, Combinatorics, Probability and Computing 7 (1998), 121-127.

PSS
J. Pach, F. Shahrokhi, and M. Szegedy: Applications of the crossing number, Algorithmica 16 (1996), 111-117.

PSo
J. Pach, J. Solymosi: Halving lines and perfect cross-matchings, in: Advances in Discrete and Computational Geometry (B. Chazelle, J.E. Goodman and R. Pollack, eds.), Contemporary Mathematics 223 AMS, Providence, 1999, 245-249.

PSoT
J. Pach, J. Solymosi, and G. Tóth: Unavoidable configurations in complete topological graphs, in: Graph Drawing 2000, Lecture Notes in Computer Science, Springer-Verlag, accepted. Also in: Discrete and Computational Geometry, submitted.

PST
J. Pach, J. Spencer, and G. Tóth: New bounds for crossing numbers, Discrete and Computational Geometry 24 (2000), 623-644.

PT
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PT0
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PT1
J. Pach and G. Tóth: Graphs drawn with few crossings per edge, Combinatorica 17 (1997), 427-439.

PT2
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PT3
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PW
J. Pach and R. Wenger: Embedding planar graphs at fixed vertex locations, in: Graph Drawing (Montréal, QC, 1998), Lecture Notes in Computer Science 1547, Springer, Berlin, 1998, 263-274.

RS
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RT
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SSS
F. Shahrokhi, O. Sýkora, L.A. Székely, and I. Vrto: The book crossing number of graphs, Journal of Graph Theory 21 (1996), 413-424.

SoT
J. Solymosi and Cs. Tóth: Distinct distances in the plane, Discrete Comput. Geom., to appear.

SW
D. Souvaine and R. Wenger: Constructing piecewise linear homeomorphisms, Comput. Geometry: Theory and Appl.

SST
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S
L. A. Székely: Crossing numbers and hard Erdos problems in discrete geometry, Combinatorics, Probability and Computing 6 (1997), 353-358.

ST1
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ST2
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T
G. Tóth: Note on geometric graphs, J. Combin. Theory Ser. A 89 (2000), 126-132,

TV
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Tr
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T1
P. Turán: A note of welcome, Journal of Graph Theory 1 (1977), 7-9.

T2
W. T. Tutte: Toward a theory of crossing numbers, Journal of Combinatorial Theory 8 (1970), 45-53.

V
P. Valtr: On geometric graphs with no k pairwise parallel edges, Disc. Comput. Geometry 19 (1998), 461-469.

W
D. R. Woodall: Thrackles and deadlock, in: Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969), Academic Press, London, 1971, 335-347