Outerstring graphs are χ-bounded

Rok, Alexandre; Walczak, Bartosz

doi:10.1145/2582112.2582115

Cited by 24 publications

(34 citation statements)

References 53 publications

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“…A class of graphs is χ-bounded if there is some function f : N → N such that, for each graph G in the class, it holds that χ(G) ≤ f (ω(G)). It is known that outer-string graphs are χ-bounded [22], while string graphs are not χ-bounded [21]. Therefore, both classes cannot be the same.…”

Section: Alternative Approachesmentioning

confidence: 99%

Refining the Hierarchies of Classes of Geometric Intersection Graphs

Cabello

Jejčič

2017

Electron. J. Combin.

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We analyse properties of geometric intersection graphs to show strict containment between some natural classes of geometric intersection graphs. In particular, we show the following properties:• A graph G is outerplanar if and only if the 1-subdivision of G is outer-segment.• For each integer k ≥ 1, the class of intersection graphs of segments with k different lengths is a strict subclass of the class of intersection graphs of segments with k + 1 different lengths.• For each integer k ≥ 1, the class of intersection graphs of disks with k different sizes is a strict subclass of the class of intersection graphs of disks with k + 1 different sizes.• The class of outer-segment graphs is a strict subclass of the class of outer-string graphs.

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Section: Alternative Approachesmentioning

confidence: 99%

Refining the Hierarchies of Classes of Geometric Intersection Graphs

Cabello

Jejčič

2017

Electron. J. Combin.

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“…Proof. It was proved in [48] that we have χ(F) = O(log n). Hence, by Theorem 1.5, χ cf (F) = O(log 3 n).…”

Section: Bounding the Cf-chromatic Number Of String Graphs In Terms Omentioning

confidence: 99%

Conflict-Free Coloring of String Graphs

Keller

Rok

Smorodinsky

2020

Discrete Comput Geom

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Conflict-free coloring (in short, CF-coloring) of a graph G = (V, E) is a coloring of V such that the neighborhood of each vertex contains a vertex whose color differs from the color of any other vertex in that neighborhood. Bounds on CF-chromatic numbers have been studied both for general graphs and for intersection graphs of geometric shapes. In this paper we obtain such bounds for several classes of string graphs, i.e., intersection graphs of curves in the plane:(i) We provide a general upper bound of O(χ(G) 2 log n) on the CF-chromatic number of any string graph G with n vertices in terms of the classical chromatic number χ(G). This result stands in contrast to general graphs where the CF-chromatic number can be Ω( √ n) already for bipartite graphs.(ii) For some central classes of string graphs, the CF-chromatic number is as large as Θ( √ n),which was shown to be the upper bound for any graph even in the non-geometric context. For several such classes (e.g., intersection graphs of frames) we prove a tight bound of Θ(log n) with respect to the relaxed notion of k-CF-coloring (in which the punctured neighborhood of each vertex contains a color that appears at most k times), for a small constant k.(iii) We obtain a general upper bound on the k-CF-chromatic number of arbitrary hypergraphs: Any hypergraph with m hyperedges can be k-CF colored withÕ(m 1 k+1 ) colors. This bound, which extends a bound of Pach and Tardos (2009), is tight for some string graphs, up to a logarithmic factor.(iv) Our fourth result concerns circle graphs in which coloring problems are motivated by VLSI designs. We prove a tight bound of Θ(log n) on the CF-chromatic number of circle graphs, and an upper bound of O(log 3 n) for a wider class of string graphs that contains circle graphs, namely, intersection graphs of grounded L-shapes.

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“…Properties of the chromatic number of geometric intersection graphs have been studied as well. For instance, Rok and Walczak proved that outer string graphs are χ-bounded [24], and Kostochka and Nešetřil [13,14] studied the chromatic number of ray graphs in terms of the girth and the clique number.…”

Section: Previous Work and Motivationmentioning

confidence: 99%

Intersection Graphs of Rays and Grounded Segments

Cardinal¹,

Felsner²,

Miltzow³

et al. 2018

JGAA

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We consider several classes of intersection graphs of line segments in the plane and prove new equality and separation results between those classes. In particular, we show that:• intersection graphs of grounded segments and intersection graphs of downward rays form the same graph class,• not every intersection graph of rays is an intersection graph of downward rays, and• not every intersection graph of rays is an outer segment graph.The first result answers an open problem posed by Cabello and Jejčič. The third result confirms a conjecture by Cabello. We thereby completely elucidate the remaining open questions on the containment relations between these classes of segment graphs. We further characterize the complexity of the recognition problems for the classes of outer segment, grounded segment, and ray intersection graphs. We prove that these recognition problems are complete for the existential theory of the reals. This holds even if a 1-string realization is given as additional input.

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Outerstring graphs are χ-bounded

Cited by 24 publications

References 53 publications

Refining the Hierarchies of Classes of Geometric Intersection Graphs

Refining the Hierarchies of Classes of Geometric Intersection Graphs

Conflict-Free Coloring of String Graphs

Intersection Graphs of Rays and Grounded Segments

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