ELSEVIER Discrete Mathematics 178 (1998) 251-255

DISCRETE MATHEMATICS

Note

On intersection representations of co-planar graphs

Jan Kratochvil*, Aleg Kub~na 1 Charles University, Prague, Czech Republic

Received 18 January 1996; revised 19 November 1996; accepted 2 December 1996

Abstract

We show that complements of planar graphs have intersection representations by convex sets in the plane, i.e., for every planar graph, one can assign convex sets in the plane to its vertices in such a way that two of the sets are disjoint if and only if the correspondning vertices are adjacent. This fact has a complexity consequence - - it follows that the problem of determining the clique number of an intersection graph of convex sets in the plane is NP-hard. We note that the complexity of this problem for intersection graphs of straight line segments in the plane is unknown. @ 1998 Elsevier Science Ltd.

Intersection graphs o f geometrical objects in the plane gain a lot of interest for their

practical applications and many nice properties. I f M is a family of subsets o f the plane

(usually the sets in M are arc-connected and determined by their geometrical shape),

we say that a graph G is an intersection graph o f M if the vertices o f G can be assigned

members o f M (say Mu E M to u E V(G)) in such a way that any two vertices u,v are

adjacent if and only if the corresponding sets Mu, My have a nonempty intersection.

In this sense interval graphs are intersection graphs o f intervals on a line, circular arc graphs are intersection graphs o f intervals on a circle, circle graphs are intersection

graphs o f chords o f a circle etc. All these classes are recognizable in polynomial time,

unlike the more general intersection graphs o f straight line segments [3, 8], intersection graphs of convex sets [8] and the most general intersection graphs o f curves (the so

called string graphs) [13,6], whose recognitions are NP-hard. (It is a challenging open

problem that for none of these three classes, the recognition is known to be in NP. Re-

sults o f Kratochvil and Matou~ek [7, 8] show that the natural approach of placing these

* The first author acknowledges partial support from Czech Research Grants GAUK 193 and 194. Mailing address: KAM MFF UK, Malostransk6 nhm. 25, 118 00 Praha 1, Czech Republic. E-mail: honza@kam.ms.mff.cuni.cz. 1 The second author is currently a student of Charles University.

0012-365X/98/$19.00 Copyright (~) 1998 Elsevier Science B.V. All rights reserved Pli S0012-365X(97)00012-5

252 J. Kratochvil, A. Kub~na/ Discrete Mathematics 178 (1998) 251-255

problems in NP by guessing representations of the input graphs necessarily fails, be- cause of the existence of graphs that require representations of exponential size.)

It is easy to show that every planar graph is a string graph. It follows from the well known "kissing lemma" (or "coin-graph theorem") of Koebe [4] that every planar graph is an intersection graph of disks in the plane (and hence also of convex sets). (For more details about the complexity of disk representations and recognition of disk intersection graphs see [11,2].) It is a long standing open problem if every planar graph is an intersection graph of straight line segments _in the plane. This problem was first formulated in 1984 by Scheinerman [14] and independently in 1990 by Pollack [personal communication]. Weaker but still open is the question if every planar graph has an intersection representation by curves such that any two curves cross at most once (Fellows [1988, personal communication]).

As concerns the 3-dimensional space, Wegner [16] showed that every graph is an intersection graph of convex sets in ~3. More recently, Thomassen [15] proved that planar graphs can be represented as touching graphs of axis-parallel graphs in the 3-space.

In this note, we pay more attention to representations of complements of planar graphs. Using the Four Color Theorem, it was observed in [5] that complements of planar graphs are intersection graphs of curves. Refining that argument we will show the following result.

Corollary I. Complements of planar 9raphs are intersection 9raphs of convex sets in the plane.

This fact has an interesting consequence for the complexity of the CLIQUE prob- lem restricted to classes of intersection graphs. It was noted in [9] that CLIQUE re- mains NP-complete for string graphs, but is polynomially solvable for intersection graphs of straight segments which are bound to follow a fixed number of direc- tions. The complexity of CLIQUE restricted to intersection graphs of segments (in unbounded number of directions) was posed there as an open problem. The strongest NP-completeness result in this direction is due to Middendorf and Pfeiffer [10] who showed that CLIQUE is NP-complete for intersection graphs of congruent L-shapes and segments, and for intersection graphs of L-shapes of two different types. Since INDEPENDENT SET is NP-complete for planar graphs, our Corollary 1 implies the following.

Corollary 2. CLIQUE is NP-complete for intersection 9raphs of convex sets in the plane.

It is not clear if one can use the same approach for the problem with segments. An affirmative solution to the following problem would indeed mean that CLIQUE is NP-complete for intersection graphs of straight line segments.

J. Kratochvil, A. KubOna/Discrete Mathematics 178 (1998) 251-255 253

vl

½

Fig. 1. Disk contact representation of G.

Problem. Is it true that complements of planar graphs are intersection graphs of straight line segments in the plane?

The rest of the paper is devoted to the proof of Corollary 1. We first prove a more general statement.

Proposition 1. Let G be a graph with vertex set partitioned into sets Vl, I12 . . . . , Vk

so that each Vi induces a complete subgraph and the 9raph G = ((1,2 . . . . ,k}, {ij " 3u E Vi, v E Vj, uv E E(G)}) is planar. Then G is an intersection 9raph o f convex sets in the plane.

Proof. We first take a system of circles Ci, i = 1,2 . . . . . k with disjoint interiors such that Ci, Cj touch each other iff ij E E(G). Such a disk contact representation of exists by the kissing lemma [4] (see an example in Fig. 1). Then we blow up the disks slightly, so that the neighboring disks intersect each other, but the intersections are relatively small. If we denote C[ the blown up circles and Aij,Bij the intersections of C/~ and Cj (for ij E E(G)), the expansion is kept small enough so that none of the segments AijBik,AijAik,BijAik,BijBik crosses the arc AijBij of C~ that lies inside C' (for

any i , j , k such that ij, ik E E(G)) .

We choose an arbitrary orientation of G and for every edge ij oriented from i to j ,

we place IV/I distinct points Diu, u E Vi, on the arc AijBij of C~ inside C[. For every u E V(G), we describe the set Nu of comers of the convex set Mu that will represent u. If u E V/ and ij E E(G) is oriented from i to j , we put Di~ in Nu. If u E Vi and ki E E(G) is oriented from k to i, we put into N~ all points Dkv such that v E Vk and uv E E(G). Finally, N, will contain the center of Ci (to guarantee that the sets Mu, u E Vi represent a complete subgraph). We set M~ -- conv(Nu), the convex hull of N~ (see an illustrative example in Fig. 2).

It is easy to show that M~,u E V(G), is an intersection representation of G. (If u E Vi, v E Vj are adjacent, then Mu M My contains either Diu o r Ojv. If u and v are nonadjacent, then Mu misses the comer Djv of My if ij is oriented from j to i (resp. My misses the comer Di~ of M~ if ij is oriented from i to j ) . No other intersections could arise.) []

254 J. Kratochvil, A. Kub#na/Discrete Mathematics 178 (1998) 251-255

c~

v

Fig. 2. Illustration to the construction of the convex representation.

By the Four Color Theorem [1, 12], every planar graph is 4-colorable. That means that if the complement of a graph G is planar then G can be partitioned into at most four cliques. For such a partition, the contracted graph G is planar (it is a subgraph of K4) and Corollary 1 follows from Proposition 1.

References

[1] K. Appel and W. Haken, Every planar map is four-colorable, Bull. Amer. Math. Soc. 82 (1976) 711-712.

[2] H. Breu, P. Hlin6n~,, D. Kirkpatrick and J. Kratochvil, The complexity of intersection and contact graphs of disks, in preparation.

[3] G. Eherlich, S. Even and R.E. Tarjan, Intersection graphs of curves in the plane, J. Combin. Theory Ser. B 21 (1976) 8-20.

[4] P. Koebe, Kontaktprobleme den konformen Abbildung, Berichte fiber die Verhandlungen der S/ichsischen Akademie der Wissenschaften, Leipzig, Math.-Physische Klasse 88 (1936) 141-164.

[5] J. Kratochvil, M. Goljan and P. Ku6era, String Graphs (Academia - - Publishing House of the Czechoslovak Academy of Sciences, Prague, 1986).

[6] J. Kratochvil, String graphs II. Recognizing string graphs is NP-hard, J. Combin. Theory Ser. B 52 (1991) 67-78.

[7] J. Kratochvil and J. Matou~ek, String graphs requiring exponential representations, J. Combin. Theory Ser. B 53 (1991) 1-4.

[8] J. Kratochvil and J. Matou~ek, Intersection graphs of segments, J. Combin. Theory Ser. B 68 (1994) 31%339.

[9] J. Kratochvil and J. Ne~e~l, INDEPENDENT SET and CLIQUE problems in intersection defined classes of graphs, Comment. Math. Univ. Carolin. 31 (1990) 85-93.

[10] M. Middendorf and F. Pfeiffer, Max clique problem in classes of string graphs, Proceedings Marseille 1989.

J. Kratochvil, A. Kub~na l Discrete Mathematics 178 (1998) 251-255 255

[11] B. Mohar, A polynomial circle packing algorithm, Discrete Math. 117 (1993) 257-263 [12] N. Roberston, D. Sanders, P. Seymour and R. Thomas, The four-colour theorem, J. Combin. Theory

Ser. B, to appear. [13] F.W. Sinden, Topology of thin film RC-circuits, Bell System Tech. J. (1966) 1639-1662. [14] E.R. Scheinerman, Intersection classes and multiple intersection parameters of graphs, Ph.D. thesis,

Princeton University, 1984. [15] C. Thomassen, Interval representations of planar graphs, J. Combin. Theory Ser. B 40 (1986) 9-20. [16] G. Wegner, Eigenschaften der Nervan Homologische-einfacher Familien im R n, Ph.D. thesis, G6ttingen,

1967.