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30
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 31 (19 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...
Pathwidth and ThreeDimensional StraightLine Grid Drawings of Graphs
"... We prove that every nvertex graph G with pathwidth pw(G) has a threedimensional straightline grid drawing with O(pw(G) n) volume. Thus for ..."
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Cited by 24 (12 self)
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We prove that every nvertex graph G with pathwidth pw(G) has a threedimensional straightline grid drawing with O(pw(G) n) volume. Thus for
ThreeDimensional Grid Drawings with SubQuadratic Volume
, 1999
"... A threedimensional grid drawing of a graph is a placement of the vertices at distinct points with integer coordinates, such that the straight linesegments representing the edges are pairwise noncrossing. A O(n volume bound is proved for threedimensional grid drawings of graphs with bounded ..."
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Cited by 18 (12 self)
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A threedimensional grid drawing of a graph is a placement of the vertices at distinct points with integer coordinates, such that the straight linesegments representing the edges are pairwise noncrossing. A O(n volume bound is proved for threedimensional grid drawings of graphs with bounded degree, graphs with bounded genus, and graphs with no bounded complete graph as a minor. The previous best bound for these graph families was O(n ). These results (partially) solve open problems due to Pach, Thiele, and Toth (1997) and Felsner, Liotta, and Wismath (2001).
The maximum number of edges in a threedimensional griddrawing
 J. Graph Algorithms Appl
, 2003
"... An exact formula is given for the maximum number of edges in a graph that admits a threedimensional griddrawing contained in a given bounding box. A threedimensional (straightline) griddrawing of a graph represents the vertices by distinct points in Z 3, and represents each edge by a linesegme ..."
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Cited by 18 (9 self)
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An exact formula is given for the maximum number of edges in a graph that admits a threedimensional griddrawing contained in a given bounding box. A threedimensional (straightline) griddrawing of a graph represents the vertices by distinct points in Z 3, and represents each edge by a linesegment between its endpoints that does not intersect any other vertex, and does not intersect any other edge except at the endpoints. A folklore result states that every (simple) graph has a threedimensional griddrawing (see [2]). We therefore are interested in griddrawings with small ‘volume’. The bounding box of a threedimensional griddrawing is the axisaligned box of minimum size that contains the drawing. By an X × Y × Z griddrawing we mean a threedimensional griddrawing, such that the edges of the bounding box contain X, Y, and Z gridpoints, respectively. The volume of a threedimensional griddrawing is the number of gridpoints in the bounding box; that is, the volume of an X ×Y ×Z griddrawing is XY Z. (This definition is formulated to ensure that a twodimensional griddrawing has positive volume.) Our main contribution is the following extremal result. Theorem 1. The maximum number of edges in an X × Y × Z griddrawing is exactly (2X − 1)(2Y − 1)(2Z − 1) − XY Z. Proof. Consider an X × Y × Z griddrawing of a graph G with n vertices and m edges. Let P be the set of points (x, y, z) contained in the bounding box such that 2x, 2y, and 2z are all integers. Observe that P  = (2X − 1)(2Y − 1)(2Z − 1). The midpoint of every edge of G is in P, and no two edges share a common midpoint. Hence m ≤ P . In addition, the midpoint of an edge does not intersect a vertex. Thus m ≤ P  − n. (1) A drawing with the maximum number of edges has no edge that passes through a gridpoint. Otherwise, subdivide the edge, and place the new vertex at that gridpoint. Thus n = XY Z, and m ≤ P  − XY Z, as claimed. This bound is attained by the following construction. Associate a vertex with each gridpoint in an X × Y × Z gridbox B. As illustrated in Figure 1, every vertex (x, y, z) is adjacent to each
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 16 (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
Queue layouts, treewidth, and threedimensional graph drawing
 Proc. 22nd Foundations of Software Technology and Theoretical Computer Science (FST TCS '02
, 2002
"... Abstract. A threedimensional (straightline grid) drawing of a graph represents the vertices by points in Z 3 and the edges by noncrossing line segments. This research is motivated by the following open problem due to Felsner, Liotta, and Wismath [Graph Drawing ’01, Lecture Notes in Comput. Sci., ..."
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Cited by 12 (6 self)
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Abstract. A threedimensional (straightline grid) drawing of a graph represents the vertices by points in Z 3 and the edges by noncrossing line segments. This research is motivated by the following open problem due to Felsner, Liotta, and Wismath [Graph Drawing ’01, Lecture Notes in Comput. Sci., 2002]: does every nvertex planar graph have a threedimensional drawing with O(n) volume? We prove that this question is almost equivalent to an existing onedimensional graph layout problem. A queue layout consists of a linear order σ of the vertices of a graph, and a partition of the edges into queues, such that no two edges in the same queue are nested with respect to σ. The minimum number of queues in a queue layout of a graph is its queuenumber. Let G be an nvertex member of a proper minorclosed family of graphs (such as a planar graph). We prove that G has a O(1) × O(1) × O(n) drawing if and only if G has O(1) queuenumber. Thus the above question is almost equivalent to an open problem of Heath, Leighton, and Rosenberg [SIAM J. Discrete Math., 1992], who ask whether every planar graph has O(1) queuenumber? We also present partial solutions to an open problem of Ganley and Heath [Discrete Appl. Math., 2001], who ask whether graphs of bounded treewidth have bounded queuenumber? We prove that graphs with bounded pathwidth, or both bounded treewidth and bounded maximum degree, have bounded queuenumber. As a corollary we obtain threedimensional drawings with optimal O(n) volume, for seriesparallel graphs, and graphs with both bounded treewidth and bounded maximum degree. 1
An Efficient Fixed Parameter Tractable Algorithm for 1Sided Crossing Minimization
 ALGORITHMICA
, 2004
"... We give an O(ϕ k · n 2) fixed parameter tractable algorithm for the 1SIDED CROSSING MINIMIZATION problem. The constant ϕ in the running time is the golden ratio ϕ = (1 + √ 5)/2 ≈ 1.618. The constant k is the parameter of the problem: the number of allowed edge crossings. ..."
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Cited by 11 (4 self)
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We give an O(ϕ k · n 2) fixed parameter tractable algorithm for the 1SIDED CROSSING MINIMIZATION problem. The constant ϕ in the running time is the golden ratio ϕ = (1 + √ 5)/2 ≈ 1.618. The constant k is the parameter of the problem: the number of allowed edge crossings.
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).
Drawing Graphs on Two and Three Lines
 GRAPH DRAWING, 10TH INTERNATIONAL SYMPOSIUM (GD 2002), VOLUME TO APPEAR OF LECTURE NOTES IN COMPUTER SCIENCE
, 2002
"... We give a lineartime algorithm to decide whether a graph has a planar LLdrawing, i.e. a planar drawingo two parallel lines. This has previo)L/ beenkno wnoLq fo trees. We utilize this resultto oult planar drawings on three lines for a generalization of bipartite graphs, also in linear time. ..."
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Cited by 7 (1 self)
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We give a lineartime algorithm to decide whether a graph has a planar LLdrawing, i.e. a planar drawingo two parallel lines. This has previo)L/ beenkno wnoLq fo trees. We utilize this resultto oult planar drawings on three lines for a generalization of bipartite graphs, also in linear time.
Pathwidth and Layered Drawings of Trees
 INTERNAT. J. COMPUT. GEOM. APPL
, 2002
"... An hlayer drawing of a graph G is a planar drawing of G in which each vertex is placed on one of h parallel lines and each edge is drawn as a straight line between its endvertices. In such a drawing, we say that an edge is proper if its endpoints lie on adjacent layers, flat if they lie on the ..."
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Cited by 7 (3 self)
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An hlayer drawing of a graph G is a planar drawing of G in which each vertex is placed on one of h parallel lines and each edge is drawn as a straight line between its endvertices. In such a drawing, we say that an edge is proper if its endpoints lie on adjacent layers, flat if they lie on the same layer and long otherwise. Thus, a proper hlayer drawing contains only proper edges, a short hlayer drawing contains no long edges, an upright hlayer drawing contains no flat edges, and an unconstrained hlayer drawing contains any type of edge. We prove