13 research outputs found

    Coloring non-crossing strings

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    For a family of geometric objects in the plane F={S1,…,Sn}\mathcal{F}=\{S_1,\ldots,S_n\}, define χ(F)\chi(\mathcal{F}) as the least integer ℓ\ell such that the elements of F\mathcal{F} can be colored with ℓ\ell colors, in such a way that any two intersecting objects have distinct colors. When F\mathcal{F} is a set of pseudo-disks that may only intersect on their boundaries, and such that any point of the plane is contained in at most kk pseudo-disks, it can be proven that χ(F)≤3k/2+o(k)\chi(\mathcal{F})\le 3k/2 + o(k) since the problem is equivalent to cyclic coloring of plane graphs. In this paper, we study the same problem when pseudo-disks are replaced by a family F\mathcal{F} of pseudo-segments (a.k.a. strings) that do not cross. In other words, any two strings of F\mathcal{F} are only allowed to "touch" each other. Such a family is said to be kk-touching if no point of the plane is contained in more than kk elements of F\mathcal{F}. We give bounds on χ(F)\chi(\mathcal{F}) as a function of kk, and in particular we show that kk-touching segments can be colored with k+5k+5 colors. This partially answers a question of Hlin\v{e}n\'y (1998) on the chromatic number of contact systems of strings.Comment: 19 pages. A preliminary version of this work appeared in the proceedings of EuroComb'09 under the title "Coloring a set of touching strings

    Contact graphs of line segments are NP-complete

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    AbstractContact graphs are a special kind of intersection graphs of geometrical objects in which the objects are not allowed to cross but only to touch each other. Contact graphs of line segments in the plane are considered — it is proved that recognizing line-segment contact graphs, with contact degrees of 3 or more, is an NP-complete problem, even for planar graphs. This result contributes to the related research on recognition complexity of curve contact graphs (Hliněný J. Combin. Theory Ser. B 74 (1998) 87)

    Characterising circular-arc contact B0B_0-VPG graphs

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    A contact B0B_0-VPG graph is a graph for which there exists a collection of nontrivial pairwise interiorly disjoint horizontal and vertical segments in one-to-one correspondence with its vertex set such that two vertices are adjacent if and only if the corresponding segments touch. It was shown by Deniz et al. that Recognition is NP\mathsf{NP}-complete for contact B0B_0-VPG graphs. In this paper we present a minimal forbidden induced subgraph characterisation of contact B0B_0-VPG graphs within the class of circular-arc graphs and provide a polynomial-time algorithm for recognising these graphs

    Subject index volumes 1–92

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    On contact graphs of paths on a grid

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    In this paper we consider Contact graphs of Paths on a Grid (CPG graphs), i.e. graphs for which there exists a family of interiorly disjoint paths on a grid in one-to-one correspondence with their vertex set such that two vertices are adjacent if and only if the corresponding paths touch at a grid-point. Our class generalizes the well studied class of VCPG graphs (see [1]). We examine CPG graphs from a structural point of view which leads to constant upper bounds on the clique number and the chromatic number. Moreover, we investigate the recognition and 3-colorability problems for B0- CPG, a subclass of CPG. We further show that CPG graphs are not necessarily planar and not all planar graphs are CPG

    Subject Index Volumes 1–200

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    Dynamic range and frequency assignment problems

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    29th International Symposium on Algorithms and Computation: ISAAC 2018, December 16-19, 2018, Jiaoxi, Yilan, Taiwan

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    Multicoloured Random Graphs: Constructions and Symmetry

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    This is a research monograph on constructions of and group actions on countable homogeneous graphs, concentrating particularly on the simple random graph and its edge-coloured variants. We study various aspects of the graphs, but the emphasis is on understanding those groups that are supported by these graphs together with links with other structures such as lattices, topologies and filters, rings and algebras, metric spaces, sets and models, Moufang loops and monoids. The large amount of background material included serves as an introduction to the theories that are used to produce the new results. The large number of references should help in making this a resource for anyone interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will appear in physic

    EUROCOMB 21 Book of extended abstracts

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