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Simultaneous embedding of planar graphs with few bends
 In 12th Symposium on Graph Drawing (GD
, 2004
"... We consider several variations of the simultaneous embedding problem for planar graphs. We begin with a simple proof that not all pairs of planar graphs have simultaneous geometric embedding. However, using bends, pairs of planar graphs can be simultaneously embedded on the O(n 2) × O(n 2) grid, wit ..."
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Cited by 36 (7 self)
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We consider several variations of the simultaneous embedding problem for planar graphs. We begin with a simple proof that not all pairs of planar graphs have simultaneous geometric embedding. However, using bends, pairs of planar graphs can be simultaneously embedded on the O(n 2) × O(n 2) grid, with at most three bends per edge, where n is the number of vertices. The O(n) time algorithm guarantees that two corresponding vertices in the graphs are mapped to the same location in the final drawing and that both the drawings are crossingfree. The special case when both input graphs are trees has several applications, such as contour tree simplification and evolutionary biology. We show that if both the input graphs are are trees, only one bend per edge is required. The O(n) time algorithm guarantees that both drawings are crossingsfree, corresponding tree vertices are mapped to the same locations, and all vertices (and bends) are on the O(n 2) × O(n 2) grid (O(n 3) × O(n 3) grid). For the special case when one of the graphs is a tree and the other is a path we can find simultaneous embedding with fixededges. That is, we can guarantee that corresponding vertices are mapped to the same locations and that corresponding edges are drawn the same way. We describe an O(n) time algorithm for simultaneous embedding with fixededges for treepath pairs with at most one bend per treeedge and no bends along path edges, such that all vertices (and bends) are on the O(n) × O(n 2) grid, (O(n 2) × O(n 3) grid).
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 36 (23 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...
Planar embeddability of the vertices of a graph using a fixed point set is NPhard
, 2003
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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 9 (1 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).
Universal Sets of n Points for Onebend Drawings of Planar Graphs with n Vertices
, 2010
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Moving Vertices to Make Drawings Plane
"... In John Tantalo’s online game Planarity the player is given a nonplane straightline drawing of a planar graph. The aim is to make the drawing plane as quickly as possible by moving vertices. In this paper we investigate the related problem MinMovedVertices which asks for the minimum number of ve ..."
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Cited by 9 (0 self)
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In John Tantalo’s online game Planarity the player is given a nonplane straightline drawing of a planar graph. The aim is to make the drawing plane as quickly as possible by moving vertices. In this paper we investigate the related problem MinMovedVertices which asks for the minimum number of vertex moves. First, we show that MinMovedVertices is NPhard and hard to approximate. Second, we establish a connection to the graphdrawing problem 1BendPointSetEmbeddability, which yields similar results for that problem. Third, we give bounds for the behavior of MinMovedVertices on trees and general planar graphs.
ManhattanGeodesic Embedding of Planar Graphs
"... In this paper, we explore a new convention for drawing graphs, the (Manhattan) geodesic drawing convention. It requires that edges are drawn as interiordisjoint monotone chains of axisparallel line segments, that is, as geodesics with respect to the Manhattan metric. First, we show that geodesic ..."
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Cited by 8 (2 self)
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In this paper, we explore a new convention for drawing graphs, the (Manhattan) geodesic drawing convention. It requires that edges are drawn as interiordisjoint monotone chains of axisparallel line segments, that is, as geodesics with respect to the Manhattan metric. First, we show that geodesic embeddability on the grid is equivalent to 1bend embeddability on the grid. For the latter question an efficient algorithm has been proposed. Second, we consider geodesic pointset embeddability where the task is to decide whether a given graph can be embedded on a given point set. We show that this problem is N Phard. In contrast, we efficiently solve geodesic polygonization—the special case where the graph is a cycle. Third, we consider geodesic pointset embeddability where the vertex–point correspondence is given. We show that on the grid, this problem is NPhard even for perfect matchings, but without the grid restriction, we solve the matching problem efficiently.