Triangle-Free Geometric Intersection Graphs with No Large Independent Sets

Discrete & Computational Geometry, Jan 2015

It is proved that there are triangle-free intersection graphs of line segments in the plane with arbitrarily small ratio between the maximum size of an independent set and the total number of vertices.

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Triangle-Free Geometric Intersection Graphs with No Large Independent Sets

Bartosz Walczak 0 0 B. Walczak Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University , Krakow , Poland It is proved that there are triangle-free intersection graphs of line segments in the plane with arbitrarily small ratio between the maximum size of an independent set and the total number of vertices. Pawlik et al. [7] proved that there are triangle-free intersection graphs of line segments in the plane with arbitrarily large chromatic number. The graphs they construct have independent sets containing more than 1/3 of all the vertices. It has been left open whether there is a constant c > 0 such that every triangle-free intersection graph of n segments in the plane has an independent set of size at least cn. Fox and Pach [3] conjectured a much more general statement, that Kk -free intersection graphs of curves in the plane have linear-size independent sets, for every k. This would imply a wellknown conjecture that k-quasi-planar graphs (graphs drawn in the plane so that no k edges cross each other) have linearly many edges [5], which is proved up to k = 4 [1]. In this note, I resolve the independent set problem in the negative, proving the following strengthening of the result of Pawlik et al.: Theorem There are triangle-free segment intersection graphs with arbitrarily small ratio between the maximum size of an independent set and the total number of vertices. - The constructions presented in the next two sections give rise to triangle-free intersection graphs of n segments in the plane with maximum independent set size ( n/ log log n). 2 Construction Pawlik et al. [7] construct, for k 1, a triangle-free graph Gk and a family Pk of subsets of V (Gk ), called probes, with the following properties: (i) |Pk | = 22k11, (ii) every member of Pk is an independent set of Gk , (iii) for every proper coloring of the vertices of Gk , there is a probe P Pk such that at least k colors are used on the vertices in P. They are built by induction on k, as follows. The graph G1 has just one vertex v, and P1 has just one probe {v}. For k 2, first, a copy (G, P) of (Gk1, Pk1) is taken. Then, for every probe P P, another copy (G P , PP ) of (Gk1, Pk1) is taken. There are no edges between vertices from different copies. Finally, for every probe P P and every probe Q PP , a new vertex dQ connected to all vertices in Q, called the diagonal of Q, is added. The resulting graph is Gk . The family of probes Pk is defined by Pk = P Q : P P and Q PP P {dQ } : P P and Q PP . It is easy to check that the graph Gk is indeed triangle-free and the conditions (i)(iii) are satisfied for (Gk , Pk )see [7] for details. It is also shown in [7] how the graph Gk is represented as a segment intersection graph. I will show that there is an assignment wk of positive integer weights to the vertices of Gk with the following properties: (iv) the total weight of Gk is k +21 22k11, (v) for every independent set I of Gk , the number of probes P Pk such that P I = is at least the weight of I . Once this is achieved, the proof of the theorem of this paper follows easily. Namely, it follows from (i) and (v) that every independent set I of Gk has weight at most 22k11. We can take the representation of Gk as a segment intersection graph and replace every segment representing a vertex v V (Gk ) by wk (v) parallel segments lying very close to each other, so as to keep the property that any two segments representing vertices u, v V (Gk ) intersect if and only if uv E (Gk ). It follows from (iv) that the family of segments obtained this way has size k +21 22k11, while every independent set of its intersection graph has size at most 22k11. The assignment wk of weights to the vertices of Gk is defined by induction on k, following the inductive construction of (Gk , Pk ). The weight of the only vertex of G1 is set to 1. This clearly satisfies (iv) and (v). For k 2, let G, P, G P , PP and dQ be defined as in the inductive step of the construction of (Gk , Pk ). Let p = |Pk1| = 22k21. The weights wk of the vertices of G are their original weights wk1 in Gk1 multiplied by p. The weights wk of the vertices of every G P are equal to their original PP (Gk1, Pk1, wk1). The proof of (iv) is straightforward: weights wk1 in Gk1. The weight wk of every diagonal dQ is set to 1. It remains to prove that (iv) and (v) are satisfied for (Gk, Pk , wk ) assuming that they hold for wk(Gk) = wk(G) + wk(G P ) + |PP | = 2 pwk1(Gk1) + p2 = k +21 22k11. For the proof of (v), let I be an independent set in Gk . Let I = {P P : P I = }. For every probe P P, define DP = {dQ : Q PP }, IP = {P PP : P I = }. By the induction hypothesis, we have wk(V (G) I ) wk(V (G P ) I ) Suppose P I. It follows that (P Q) I = and (P {dQ}) I = for every Q PP . Hence |IP | = |PP | = 2 p. Moreover, we have dQ / I whenever Q IP , because dQ is connected to all vertices in Q, one of which belongs to I . Hence wk(V (G P ) I ) + wk(DP I ) IP | = |PP | = p. Now, suppose P P I. If Q IP , then (P Q) I = , dQ / I (by the same argument as above), and (P {dQ}) I = . If Q PP IP , then (P Q) I = , and (P {dQ}) I = if and only if dQ I . Hence wk(V (G P ) I ) + wk(DP I ) |IP | + |DP I | = |IP |. To conclude, we gather all the inequalities and obtain wk(I ) = wk(V (G) I ) + wk(V (G P ) I ) + wk(DP I ) PP PI PP I |IP | = PI |IP | = PP I PP 3 Improved Construction Pawlik et al. [7] define also a graph G k , which arises from (Gk, Pk ) by adding, for every probe P Pk, a diagonal dP connected to all vertices in P. This is the smallest triangle-free segment intersection graph known to have chromatic number greater than k. Define the assignment w k of weights to the vertices of G k so that w k is equal to wk on the vertices of Gk and w k(dP ) = 1 for every P Pk . Let I be an independent set in G k. Let I = {P Pk : P I = }. Hence dP / I for P I. It follows that |I| + |Pk I| = |Pk | = 2 w k (G k ) = wk (Gk ) + |Pk | = k +23 22k11. The graph G k is the smallest one for which I can prove that it has a weight assignment such that the ratio between the maximum weight of an independent set and the total weight is at most k +23 . It is not difficult to prove (e.g. using weak LP duality) that the assignment of weights w k to the vertices of G k is optimal (gives the least ratio) for this particular graph. Both constructions give rise to triangle-free intersection graphs of n segments in the plane with maximum independent set size ( n/ log log n). On the other hand, it follows from the result of McGuinness [4] that every triangle-free intersection graph of n segments has chromatic number O (log n) and maximum independent set size ( n/ log n). 4 Other Geometric Shapes It is known that the graphs Gk and G k have intersection models by many other geometric shapes, for example, L-shapes, axis-parallel ellipses, circles, axis-parallel square boundaries [6] or axis-parallel boxes in R3 [2]. The result of this paper can be extended to those models for which every geometric object X representing a vertex of the intersection graph can be replaced by many pairwise disjoint objects approximating X . This is possible, for example, for intersection graphs of L-shapes, circles or axisparallel square boundaries, but not for intersection graphs of axis-parallel ellipses or axis-parallel boxes in R3. The problem whether triangle-free intersection graphs of the latter kind of shapes have linear-size independent sets remains open. Acknowledgments The author thanks Michael Hoffmann for helpful discussions. The work was supported by Polish National Science Center Grant 2011/03/B/ST6/01367 and Swiss National Science Foundation Grant 200020-144531. Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.


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Triangle-Free Geometric Intersection Graphs with No Large Independent Sets, Discrete & Computational Geometry, 2015, 221-225, DOI: 10.1007/s00454-014-9645-y