String graphs and separators
نویسنده
چکیده
String graphs, that is, intersection graphs of curves in the plane, have been studied since the 1960s. We provide an expository presentation of several results, including very recent ones: some string graphs require an exponential number of crossings in every string representation; exponential number is always sufficient; string graphs have small separators; and the current best bound on the crossing number of a graph in terms of paircrossing number. For the existence of small separators, the proof includes generally useful results on approximate flow-cut dualities. This expository paper was prepared as a material for two courses co-taught by the author in 2013, at Charles University and at ETH Zurich. It aims at a complete and streamlined presentation of several results concerning string graphs. This important and challenging class of intersection graphs has traditionally been studied at the Department of Applied Mathematics of the Charles University, especially by Jan Kratochv́ıl and his students and collaborators. A major part of the paper is devoted to a separator theorem by Fox and Pach, recently improved by the author, as well as an application of it by Tóth in a challenging problem from graph drawing, namely, bounding the crossing number by a function of the pair-crossing number. This is an excellent example of a mathematical proof with a simple idea but relying on a number of other results from different areas. The proof is presented in full, assuming very little as a foundation, so that the reader can see everything that is involved. A key step is an approximate flow-cut duality from combinatorial optimization and approximation algorithms, whose proof relies on linear programming duality and a theorem on metric embeddings. Acknowledgments. I am very grateful to Rado Fulek, Vincent Kusters, Jan Kynčl, and Zuzana Safernová for proofreading, comments, and corrections. It was a pleasure to teach the courses together with Pavel Valtr in Prague and with Michael Hoffmann and Emo Welzl in Zurich, and to work with Jan Kratochv́ıl on questions in string graphs as well as on many other things. I also thank an anonymous referee for numerous useful remarks and suggestions. ∗Supported by the ERC Advanced Grant No. 267165 and by the project CE-ITI (GACR P202/12/G061). 1 ar X iv :1 31 1. 50 48 v2 [ m at h. C O ] 2 5 A pr 2 01 4 1 Intersection graphs The classes IG(M). Let M be a system of sets; we will typically consider systems of geometrically defined subsets of R2, such as all segments in the plane. We define IG(M), the class of intersection graphs of M, by IG(M) = { (V,E) : V ⊆M, E = {{M,M ′} ∈ ( V 2 ) : M ∩M ′ 6= ∅} } . In words, the vertices of each graph in IG(M) are sets in M, and two vertices are connected by an edge if they have a nonempty intersection. Usually we consider intersection graphs of M up to isomorphism; i.e., we regard a graph G as an intersection graph of M if it is merely isomorphic to a graph G′ ∈ IG(M). In that case we call V (G′) ⊆ M an M-representation of G, or just a representation of G if M is understood. Important examples. • For M consisting of all (closed) intervals on the real line, we obtain the class of interval graphs. This is one of the most useful graph classes in applications. Interval graphs have several characterizations, they can be recognized in linear time, and there is even a detective story Who Killed the Duke of Densmore? by Claude Berge (in French; see [Ber95] for English translation) in which the solution depends on properties of interval graphs. • Disk graphs, i.e., intersection graphs of disks in the plane, and unit disk graphs have been studied extensively. Of course, one can also investigate intersection graphs of balls in Rd for a given d, or of unit balls. • Another interesting class is CONV, the intersection graphs of convex sets in the plane. • Here we will devote most of the time to the class STRING of string graphs, the intersection graphs of simple curves in the plane. • Another important class is SEG, the segment graphs, which are the intersection graphs of line segments in R2. Other interesting classes of graphs are obtained by placing various restrictions on the mutual position or intersection pattern of the sets representing the vertices. For example: • For an integer k ≥ 1, k-STRING is the subclass of string graphs consisting of all graphs representable by curves such that every two of them have at most k points of intersection.1 Some authors moreover require that each of the intersection points is a crossing, i.e., a point where, locally, one of the edges passes from one side of the second edge to the other (as opposed to a touching point).
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