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Drawings of planar graphs with few slopes and segments
 Computational Geometry Theory and Applications 38:194–212
, 2005
"... We study straightline drawings of planar graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2trees, and planar 3trees. We prove that every 3connected plane graph on n vertices has a plane drawing with at most 5 2 ..."
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Cited by 18 (5 self)
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We study straightline drawings of planar graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2trees, and planar 3trees. We prove that every 3connected plane graph on n vertices has a plane drawing with at most 5 2n segments and at most 2n slopes. We prove that every cubic 3connected plane graph has a plane drawing with three slopes (and three bends on the outerface). In a companion paper, drawings of nonplanar graphs with few slopes are also considered.
Sublogarithmic Distributed MIS Algorithm for Sparse Graphs using NashWilliams Decomposition
 In Journal of Distributed Computing Special Issue of selected papers from PODC
, 2008
"... We study the distributed maximal independent set (henceforth, MIS) problem on sparse graphs. Currently, there are known algorithms with a sublogarithmic running time for this problem on oriented trees and graphs of bounded degrees. We devise the first sublogarithmic algorithm for computing MIS on gr ..."
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Cited by 15 (2 self)
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We study the distributed maximal independent set (henceforth, MIS) problem on sparse graphs. Currently, there are known algorithms with a sublogarithmic running time for this problem on oriented trees and graphs of bounded degrees. We devise the first sublogarithmic algorithm for computing MIS on graphs of bounded arboricity. This is a large family of graphs that includes graphs of bounded degree, planar graphs, graphs of bounded genus, graphs of bounded treewidth, graphs that exclude a fixed minor, and many other graphs. We also devise efficient algorithms for coloring graphs from these families. These results are achieved by the following technique that may be of independent interest. Our algorithm starts with computing a certain graphtheoretic structure, called NashWilliams forestsdecomposition. Then this structure is used to compute the MIS or coloring. Our results demonstrate that this methodology is very powerful. Finally, we show nearlytight lower bounds on the running time of any distributed algorithm for computing a forestsdecomposition.
Boundeddegree graphs have arbitrarily large geometric thickness
 Electron. J. Combin
"... Abstract. The geometric thickness of a graph G is the minimum integer k such that there is a straight line drawing of G with its edge set partitioned into k plane subgraphs. Eppstein [Separating thickness from geometric thickness. In Towards a Theory of Geometric Graphs, vol. 342 of Contemp. Math., ..."
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Cited by 14 (6 self)
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Abstract. The geometric thickness of a graph G is the minimum integer k such that there is a straight line drawing of G with its edge set partitioned into k plane subgraphs. Eppstein [Separating thickness from geometric thickness. In Towards a Theory of Geometric Graphs, vol. 342 of Contemp. Math., AMS, 2004] asked whether every graph of bounded maximum degree has bounded geometric thickness. We answer this question in the negative, by proving that there exists ∆regular graphs with arbitrarily large geometric thickness. In particular, for all ∆ ≥ 9 and for all large n, there exists a ∆regular graph with geometric thickness at least c √ ∆n 1/2−4/∆−ǫ. Analogous results concerning graph drawings with few edge slopes are also presented, thus solving open problems by Dujmović et al. [Really straight graph drawings. In Proc. 12th
A polynomial bound for untangling geometric planar graphs
, 2007
"... ABSTRACT. To untangle a geometric graph means to move some of the vertices so that the resulting geometric graph has no crossings. Pach and Tardos [Discrete Comput. Geom., 2002] asked if every nvertex geometric planar graph can be untangled while keeping at least n ɛ vertices fixed. We answer this ..."
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Cited by 13 (1 self)
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ABSTRACT. To untangle a geometric graph means to move some of the vertices so that the resulting geometric graph has no crossings. Pach and Tardos [Discrete Comput. Geom., 2002] asked if every nvertex geometric planar graph can be untangled while keeping at least n ɛ vertices fixed. We answer this question in the affirmative with ɛ = 1/4. The previous best known bound was Ω ( � log n / log log n). We also consider untangling geometric trees. It is known that every nvertex geometric tree can be untangled while keeping at least � n/3 vertices fixed, while the best upper bound was O((n log n) 2/3). We answer a question of Spillner and Wolff
Partitions of Complete Geometric Graphs into Plane Trees
 IN 12TH INTERNATIONAL SYMPOSIUM ON GRAPH DRAWING (GD ’04
, 2006
"... Consider the open problem: does every complete geometric graph K 2n have a partition of its edge set into n plane spanning trees? We approach this problem from three directions. First, we study the case of convex geometric graphs. It is well known that the complete convex graph K 2n has a partition ..."
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Cited by 6 (2 self)
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Consider the open problem: does every complete geometric graph K 2n have a partition of its edge set into n plane spanning trees? We approach this problem from three directions. First, we study the case of convex geometric graphs. It is well known that the complete convex graph K 2n has a partition into n plane spanning trees. We characterise all such partitions. Second, we give a sufficient condition, which generalises the convex case, for a complete geometric graph to have a partition into plane spanning trees. Finally, we consider a relaxation of the problem in which the trees of the partition are not necessarily spanning. We prove that every complete geometric graph Kn can be partitioned into at most n n/12 plane trees.
Compact Navigation and Distance Oracles for Graphs with Small Treewidth ⋆
"... Abstract. Given an unlabeled, unweighted, and undirected graph with n vertices and small (but not necessarily constant) treewidth k, we consider the problem of preprocessing the graph to build spaceefficient encodings (oracles) to perform various queries efficiently. We assume the word RAM model wh ..."
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Cited by 2 (0 self)
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Abstract. Given an unlabeled, unweighted, and undirected graph with n vertices and small (but not necessarily constant) treewidth k, we consider the problem of preprocessing the graph to build spaceefficient encodings (oracles) to perform various queries efficiently. We assume the word RAM model where the size of a word is Ω (log n) bits. The first oracle, we present, is the navigation oracle which facilitates primitive navigation operations of adjacency, neighborhood, and degree queries. By way of an enumerate argument, which is of independent interest, we show the space requirement of the oracle is optimal to within lower order terms for all treewidths. The oracle supports the mentioned queries all in constant worstcase time. The second oracle, we present, is an exact distance oracle which facilitates distance queries between any pair of vertices (i.e., an allpair shortestpath oracle). The space requirement of the oracle is also optimal to within lower order terms. Moreover, the distance queries perform in O ( k 2 log 3 k) time. Particularly, for the class of graphs of our interest, graphs of bounded treewidth (where k is constant), the distances are reported in constant worstcase time. 1
Really straight drawings I: Planar graphs
, 2005
"... We study straightline drawings of planar graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2trees, and planar 3trees. We prove that every 3connected plane graph on n vertices has a plane drawing with at most 5n/ ..."
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Cited by 1 (1 self)
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We study straightline drawings of planar graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2trees, and planar 3trees. We prove that every 3connected plane graph on n vertices has a plane drawing with at most 5n/2 segments and at most 2n slopes. We prove that every cubic 3connected plane graph has a plane drawing with three slopes (and three bends on the outerface). In a companion paper, drawings of nonplanar graphs with few slopes are also considered.
Cubic graphs have bounded slope parameter
, 2009
"... We show that every finite connected graph G with maximum degree three and with at least one vertex of degree smaller than three has a straightline drawing in the plane satisfying the following conditions. No three vertices are collinear, and a pair of vertices form an edge in G if and only if the s ..."
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We show that every finite connected graph G with maximum degree three and with at least one vertex of degree smaller than three has a straightline drawing in the plane satisfying the following conditions. No three vertices are collinear, and a pair of vertices form an edge in G if and only if the segment connecting them is parallel to one of the sides of a previously fixed regular pentagon. It is also proved that every finite graph with maximum degree three permits a straightline drawing with the above properties using at most seven different edge slopes. Submitted:
On the book thickness of ktrees ∗
, 911
"... Every ktree has book thickness at most k + 1, and this bound is best possible for all k ≥ 3. Vandenbussche et al. (2009) proved that every ktree that has a smooth degree3 tree decomposition with width k has book thickness at most k. We prove this result is best possible for k ≥ 4, by constructing ..."
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Every ktree has book thickness at most k + 1, and this bound is best possible for all k ≥ 3. Vandenbussche et al. (2009) proved that every ktree that has a smooth degree3 tree decomposition with width k has book thickness at most k. We prove this result is best possible for k ≥ 4, by constructing a ktree with book thickness k + 1 that has a smooth degree4 tree decomposition with width k. This solves an open problem of Vandenbussche et al. (2009) 1