Bounded-Degree Independent Sets in Planar Graphs

Biedl, Thérèse; Wilkinson, Dana

doi:10.1007/s00224-005-1139-0

Cited by 7 publications

(11 citation statements)

References 11 publications

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“…• The proof of Theorem 5 is similar to a strategy developed by Biedl and Wilkinson [1] for finding bounded degree independent sets in planar graphs.…”

Section: The Results Followsmentioning

confidence: 99%

“…Motivated by applications in computational geometry, the previously known results regarding bounded degree independent sets have been for planar graphs [1,5,6,8]. The best results were obtained by Biedl and Wilkinson [1], who proved tight bounds (up to an additive constant) on α d (G) for planar G with d ≤ 15. For d ≥ 16 there is a gap in the bounds.…”

Section: Bounded Degree Independent Setsmentioning

confidence: 99%

See 1 more Smart Citation

Induced Subgraphs of Bounded Degree and Bounded Treewidth

Bose

Dujmović

Wood

2005

Lecture Notes in Computer Science

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We prove that for all 0 ≤ t ≤ k and d ≥ 2k, every graph G with treewidth at most k has a 'large' induced subgraph H, where H has treewidth at most t and every vertex in H has degree at most d in G. The order of H depends on t, k, d, and the order of G. With t = k, we obtain large sets of bounded degree vertices. With t = 0, we obtain large independent sets of bounded degree. In both these cases, our bounds on the order of H are tight. For bounded degree independent sets in trees, we characterise the extremal graphs. Finally, we prove that an interval graph with maximum clique size k has a maximum independent set in which every vertex has degree at most 2k.

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“…• The proof of Theorem 5 is similar to a strategy developed by Biedl and Wilkinson [1] for finding bounded degree independent sets in planar graphs.…”

Section: The Results Followsmentioning

confidence: 99%

Section: Bounded Degree Independent Setsmentioning

confidence: 99%

Induced Subgraphs of Bounded Degree and Bounded Treewidth

Bose

Dujmović

Wood

2005

Lecture Notes in Computer Science

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“…Dobkin and Kirkpatrick [9] showed how to construct a DK-hierarchy. This construction was later improved by Biedl and Wilkinson [1]. Formally, they start by defining P 1 = P .…”

Section: Polyhedra In 3d Spacementioning

confidence: 99%

“…Since we removed at least 1/10-th of the vertices after each iteration of Case 1 [1], after log 2 (log 3 c) rounds the size of the current polyhedron is at most (9/10) log 2 (log 3 c) |P i |. At this point, we run into Case 2 and add extra vertices to the polyhedron.…”

Section: Polyhedra In 3d Spacementioning

confidence: 99%

Optimal detection of intersections between convex polyhedra

Barba

Langerman

2014

Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms

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For a polyhedron P in R d , denote by |P | its combinatorial complexity, i.e., the number of faces of all dimensions of the polyhedra. In this paper, we revisit the classic problem of preprocessing polyhedra independently so that given two preprocessed polyhedra P and Q in R d , each translated and rotated, their intersection can be tested rapidly.For d = 3 we show how to perform such a test in O(log |P | + log |Q|) time after linear preprocessing time and space. This running time is the best possible and improves upon the last best known query time of O(log |P | log |Q|) by Dobkin and Kirkpatrick (1990).We then generalize our method to any constant dimension d, achieving the same optimal O(log |P |+log |Q|) query time using a representation of size O(|P | d/2 +ε ) for any ε > 0 arbitrarily small. This answers an even older question posed by Dobkin and Kirkpatrick 30 years ago.In addition, we provide an alternative O(log |P | + log |Q|) algorithm to test the intersection of two convex polygons P and Q in the plane. IntroductionConstructing or detecting the intersection between geometric objects has been an important subject of study in computational geometry. It was one of the main questions addressed in Shamos' seminal paper that lay the grounds of computational geometry [23], the first application of the plane sweep technique [24], and is still the topic of several volumes being published today.Extensive research has focused on finding efficient algorithms for intersection testing or collision detection as this class of problems has countless applications in motion planning, robotics, computer graphics, Computer-Aided Design, VLSI design and more. For information on collision detection refer to surveys [16,17] and to Chapter 38 of the Handbook of Computational Geometry [15].The first problem to be addressed is to compute the intersection of two convex objects. In this paper we focus on convex polygons and convex polyhedra (or simply polyhedra). Let P and Q be two polyhedra to be tested for intersection. Let |P | and |Q| denote the combinatorial complexities of P and Q, respectively, i.e., the number of faces of all dimensions of the polygon or polyhedra (vertices are 0-dimensional faces while edges are 1-dimensional faces). Let n = |P | + |Q| denote the total complexity.In the plane, Shamos [23] presented an optimal Θ(n)-time algorithm to construct the intersection of a pair of convex polygons. Another linear time algorithm was later presented by O'Rourke et al. [22]. In 3D space, Muller and Preparata [20] proposed an O(n log n) time algorithm to test whether two polyhedra in three-dimensional space intersect. Their algorithm has a second phase which computes the intersection of these polyhedra within the same running time using geometric dualization. Dobkin and Kirkpatrick [9] introduced a hierarchical data structure to represent a polyhedron that allows them to test if two polyhedra intersect in linear time. In a subsequent paper, Chazelle [2] presented an optimal linear time algorithm to compute the...

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“…We mention in passing that related papers were written (that came from distinct inspiration), see e.g. [1,[6][7][8]12], from which some lower bounds on the regular k-independence number can be obtained, but which are inferior to the lower bound obtained by the current approach.…”

Section: Introduction and Benchmark Boundsmentioning

confidence: 99%

Regular independent sets

Caro

Hansberg²,

Pepper

2016

Discrete Applied Mathematics

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The regular independence number, introduced by Albertson and Boutin in 1990, is the size of a largest set of independent vertices with the same degree. Lower bounds were proven for this invariant, in terms of the order, for trees and planar graphs. In this article, we generalize and extend these results to find lower bounds for the regular k-independence number for trees, forests, planar graphs, k-trees and k-degenerate graphs.

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Bounded-Degree Independent Sets in Planar Graphs

Cited by 7 publications

References 11 publications

Induced Subgraphs of Bounded Degree and Bounded Treewidth

Induced Subgraphs of Bounded Degree and Bounded Treewidth

Optimal detection of intersections between convex polyhedra

Regular independent sets

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