17,271 research outputs found

    Integer realizations of disk and segment graphs

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    A disk graph is the intersection graph of disks in the plane, a unit disk graph is the intersection graph of same radius disks in the plane, and a segment graph is an intersection graph of line segments in the plane. It can be seen that every disk graph can be realized by disks with centers on the integer grid and with integer radii; and similarly every unit disk graph can be realized by disks with centers on the integer grid and equal (integer) radius; and every segment graph can be realized by segments whose endpoints lie on the integer grid. Here we show that there exist disk graphs on nn vertices such that in every realization by integer disks at least one coordinate or radius is 22Ω(n)2^{2^{\Omega(n)}} and on the other hand every disk graph can be realized by disks with integer coordinates and radii that are at most 22O(n)2^{2^{O(n)}}; and we show the analogous results for unit disk graphs and segment graphs. For (unit) disk graphs this answers a question of Spinrad, and for segment graphs this improves over a previous result by Kratochv\'{\i}l and Matou{\v{s}}ek.Comment: 35 pages, 14 figures, corrected a typ

    Bidimensionality of Geometric Intersection Graphs

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    Let B be a finite collection of geometric (not necessarily convex) bodies in the plane. Clearly, this class of geometric objects naturally generalizes the class of disks, lines, ellipsoids, and even convex polygons. We consider geometric intersection graphs GB where each body of the collection B is represented by a vertex, and two vertices of GB are adjacent if the intersection of the corresponding bodies is non-empty. For such graph classes and under natural restrictions on their maximum degree or subgraph exclusion, we prove that the relation between their treewidth and the maximum size of a grid minor is linear. These combinatorial results vastly extend the applicability of all the meta-algorithmic results of the bidimensionality theory to geometrically defined graph classes

    Spanners for Geometric Intersection Graphs

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    Efficient algorithms are presented for constructing spanners in geometric intersection graphs. For a unit ball graph in R^k, a (1+\epsilon)-spanner is obtained using efficient partitioning of the space into hypercubes and solving bichromatic closest pair problems. The spanner construction has almost equivalent complexity to the construction of Euclidean minimum spanning trees. The results are extended to arbitrary ball graphs with a sub-quadratic running time. For unit ball graphs, the spanners have a small separator decomposition which can be used to obtain efficient algorithms for approximating proximity problems like diameter and distance queries. The results on compressed quadtrees, geometric graph separators, and diameter approximation might be of independent interest.Comment: 16 pages, 5 figures, Late

    Hyperbolic intersection graphs and (quasi)-polynomial time

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    We study unit ball graphs (and, more generally, so-called noisy uniform ball graphs) in dd-dimensional hyperbolic space, which we denote by Hd\mathbb{H}^d. Using a new separator theorem, we show that unit ball graphs in Hd\mathbb{H}^d enjoy similar properties as their Euclidean counterparts, but in one dimension lower: many standard graph problems, such as Independent Set, Dominating Set, Steiner Tree, and Hamiltonian Cycle can be solved in 2O(n11/(d1))2^{O(n^{1-1/(d-1)})} time for any fixed d3d\geq 3, while the same problems need 2O(n11/d)2^{O(n^{1-1/d})} time in Rd\mathbb{R}^d. We also show that these algorithms in Hd\mathbb{H}^d are optimal up to constant factors in the exponent under ETH. This drop in dimension has the largest impact in H2\mathbb{H}^2, where we introduce a new technique to bound the treewidth of noisy uniform disk graphs. The bounds yield quasi-polynomial (nO(logn)n^{O(\log n)}) algorithms for all of the studied problems, while in the case of Hamiltonian Cycle and 33-Coloring we even get polynomial time algorithms. Furthermore, if the underlying noisy disks in H2\mathbb{H}^2 have constant maximum degree, then all studied problems can be solved in polynomial time. This contrasts with the fact that these problems require 2Ω(n)2^{\Omega(\sqrt{n})} time under ETH in constant maximum degree Euclidean unit disk graphs. Finally, we complement our quasi-polynomial algorithm for Independent Set in noisy uniform disk graphs with a matching nΩ(logn)n^{\Omega(\log n)} lower bound under ETH. This shows that the hyperbolic plane is a potential source of NP-intermediate problems.Comment: Short version appears in SODA 202

    Graph Isomorphism for unit square graphs

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    In the past decades for more and more graph classes the Graph Isomorphism Problem was shown to be solvable in polynomial time. An interesting family of graph classes arises from intersection graphs of geometric objects. In this work we show that the Graph Isomorphism Problem for unit square graphs, intersection graphs of axis-parallel unit squares in the plane, can be solved in polynomial time. Since the recognition problem for this class of graphs is NP-hard we can not rely on standard techniques for geometric graphs based on constructing a canonical realization. Instead, we develop new techniques which combine structural insights into the class of unit square graphs with understanding of the automorphism group of such graphs. For the latter we introduce a generalization of bounded degree graphs which is used to capture the main structure of unit square graphs. Using group theoretic algorithms we obtain sufficient information to solve the isomorphism problem for unit square graphs.Comment: 31 pages, 6 figure
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