Results 1  10
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13
On Linear Layouts of Graphs
, 2004
"... In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A kstack (resp... ..."
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Cited by 30 (18 self)
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In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A kstack (resp...
Treepartitions of ktrees with applications in graph layout
 Proc. 29th Workshop on Graph Theoretic Concepts in Computer Science (WG’03
, 2002
"... Abstract. A treepartition of a graph is a partition of its vertices into ‘bags ’ such that contracting each bag into a single vertex gives a forest. It is proved that every ktree has a treepartition such that each bag induces a (k − 1)tree, amongst other properties. Applications of this result t ..."
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Cited by 15 (11 self)
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Abstract. A treepartition of a graph is a partition of its vertices into ‘bags ’ such that contracting each bag into a single vertex gives a forest. It is proved that every ktree has a treepartition such that each bag induces a (k − 1)tree, amongst other properties. Applications of this result to two wellstudied models of graph layout are presented. First it is proved that graphs of bounded treewidth have bounded queuenumber, thus resolving an open problem due to Ganley and Heath [2001] and disproving a conjecture of Pemmaraju [1992]. This result provides renewed hope for the positive resolution of a number of open problems regarding queue layouts. In a related result, it is proved that graphs of bounded treewidth have threedimensional straightline grid drawings with linear volume, which represents the largest known class of graphs with such drawings. 1
Drawing Kn in Three Dimensions with One Bend per Edge
, 2006
"... We give a drawing of Kn in three dimensions in which vertices are placed at integer grid points and edges are drawn crossingfree with at most one bend per edge in a volume bounded by O(n^2.5). ..."
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Cited by 8 (0 self)
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We give a drawing of Kn in three dimensions in which vertices are placed at integer grid points and edges are drawn crossingfree with at most one bend per edge in a volume bounded by O(n^2.5).
ThreeDimensional 1Bend Graph Drawings
 Concordia University
, 2004
"... We consider threedimensional griddrawings of graphs with at most one bend per edge. Under the additional requirement that the vertices be collinear, we prove that the minimum volume of such a drawing is Θ(cn), where n is the number of vertices and c is the cutwidth of the graph. We then prove that ..."
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Cited by 4 (0 self)
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We consider threedimensional griddrawings of graphs with at most one bend per edge. Under the additional requirement that the vertices be collinear, we prove that the minimum volume of such a drawing is Θ(cn), where n is the number of vertices and c is the cutwidth of the graph. We then prove that every graph has a threedimensional griddrawing with O(n 3 / log 2 n) volume and one bend per edge. The best previous bound was O(n 3).
On the Queue Number of Planar Graphs
, 2010
"... We prove that planar graphs have O(log 4 n) queue number, thus improving upon the previous O ( √ n) upper bound. Consequently, planar graphs admit 3D straightline crossingfree grid drawings in O(n log c n) volume, for some constant c, thus improving upon the previous O(n 3/2) upper bound. 2 1 ..."
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Cited by 3 (0 self)
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We prove that planar graphs have O(log 4 n) queue number, thus improving upon the previous O ( √ n) upper bound. Consequently, planar graphs admit 3D straightline crossingfree grid drawings in O(n log c n) volume, for some constant c, thus improving upon the previous O(n 3/2) upper bound. 2 1
Nothreeinlinein3D
 In Proc. 12th Int. Symp. on Graph Drawing (GD’04) [GD004
, 2004
"... The nothreeinline problem, introduced by Dudeney in 1917, asks for the maximum number of points in the nn grid with no three points collinear. In 1951, Erdos proved that the answer is (n). We consider the analogous threedimensional problem, and prove that the maximum number of points in the ..."
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Cited by 1 (0 self)
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The nothreeinline problem, introduced by Dudeney in 1917, asks for the maximum number of points in the nn grid with no three points collinear. In 1951, Erdos proved that the answer is (n). We consider the analogous threedimensional problem, and prove that the maximum number of points in the n n n grid with no three collinear is (n ).
Boundeddegree graphs have arbitrarily large geometric thickness
 Electron. J. Combin
, 509
"... It is proved that there exist graphs of bounded degree with arbitrarily large queuenumber. In particular, for all ∆ ≥ 3 and for all sufficiently large n, there is a simple ∆regular nvertex graph with queuenumber at least c √ ∆n 1/2−1/∆ for some absolute constant c. ..."
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Cited by 1 (1 self)
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It is proved that there exist graphs of bounded degree with arbitrarily large queuenumber. In particular, for all ∆ ≥ 3 and for all sufficiently large n, there is a simple ∆regular nvertex graph with queuenumber at least c √ ∆n 1/2−1/∆ for some absolute constant c.
Upward threedimensional grid drawings of graphs. arXiv.org math.CO/0510051
, 2005
"... Abstract. A threedimensional grid drawing of a graph is a placement of the vertices at distinct points with integer coordinates, such that the straight line segments representing the edges do not cross. Our aim is to produce threedimensional grid drawings with small bounding box volume. Our first ..."
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Cited by 1 (1 self)
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Abstract. A threedimensional grid drawing of a graph is a placement of the vertices at distinct points with integer coordinates, such that the straight line segments representing the edges do not cross. Our aim is to produce threedimensional grid drawings with small bounding box volume. Our first main result is that every nvertex graph with bounded degeneracy has a threedimensional grid drawing with O(n 3/2) volume. This is the largest known class of graphs that have such drawings. A threedimensional grid drawing of a directed acyclic graph (dag) is upward if every arc points up in the zdirection. We prove that every dag has an upward threedimensional grid drawing with O(n 3) volume, which is tight for the complete dag. The previous best upper bound was O(n 4). Our main result concerning upward drawings is that every ccolourable dag (c constant) has an upward threedimensional grid drawing with O(n 2) volume. This result matches the bound in the undirected case, and improves the best known bound from O(n 3) for many classes of dags, including planar, series parallel, and outerplanar. Improved bounds are also obtained for tree dags. We prove a strong relationship between upward threedimensional grid drawings, upward track layouts, and upward queue layouts. Finally, we study upward threedimensional grid drawings with bends in the edges. 1.
Induced matchings and induced paths in graphs
 UNPUBLISHED MANUSCRIPT, KAMDIMATIA SERIES
, 2007
"... Denote by ∇1(G) the maximum of E(H) over all (simple) minors V (H) of G obtained by contracting a star forest. We prove that there exists a positive function ǫ such that every graph G of order n has (at least) two clones (that is two vertices with the same neighbours) or an induced matching of ..."
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Denote by ∇1(G) the maximum of E(H) over all (simple) minors V (H) of G obtained by contracting a star forest. We prove that there exists a positive function ǫ such that every graph G of order n has (at least) two clones (that is two vertices with the same neighbours) or an induced matching of size at least ǫ(∇1(G))n and that this set may be found in linear time. More generally, we prove that for every integer k there exists a (very slowly growing) positive function ǫk such that every graph of order n has an involutive automorphism or includes a set of size at least k⌊ǫk(∇⌊k/2⌋(G))n ⌋ inducing ⌊ǫk(∇⌊k/2⌋(G))n ⌋ disjoint paths on k vertices.