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Drawing Planar Graphs Using the Canonical Ordering
 ALGORITHMICA
, 1996
"... We introduce a new method to optimize the required area, minimum angle and number of bends of planar drawings of graphs on a grid. The main tool is a new type of ordering on the vertices and faces of triconnected planar graphs. Using this method linear time and space algorithms can be designed for m ..."
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Cited by 65 (0 self)
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We introduce a new method to optimize the required area, minimum angle and number of bends of planar drawings of graphs on a grid. The main tool is a new type of ordering on the vertices and faces of triconnected planar graphs. Using this method linear time and space algorithms can be designed for many graph drawing problems.  Every triconnected planar graph G can be drawn convexly with straight lines on an (2n \Gamma 4) \Theta (n \Gamma 2) grid, where n is the number of vertices.  Every triconnected planar graph with maximum degree four can be drawn orthogonally on an n \Theta n grid with at most d 3n 2 e + 4, and if n ? 6 then every edge has at most two bends.  Every 3planar graph G can be drawn with at most b n 2 c + 1 bends on an b n 2 c \Theta b n 2 c grid.  Every triconnected planar graph G can be drawn planar on an (2n \Gamma 6) \Theta (3n \Gamma 9) grid with minimum angle larger than 2 d radians and at most 5n \Gamma 15 bends, with d the maximum d...
ONLINE PLANARITY TESTING
, 1996
"... The online planaritytesting problem consists of performing the following operations on a planar graph G: (i) testing if a new edge can be added to G so that the resulting graph is itself planar; (ii) adding vertices and edges such that planarity is preserved. An efficient technique for online plan ..."
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Cited by 24 (2 self)
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The online planaritytesting problem consists of performing the following operations on a planar graph G: (i) testing if a new edge can be added to G so that the resulting graph is itself planar; (ii) adding vertices and edges such that planarity is preserved. An efficient technique for online planarity testing of a graph is presented that uses O(n) space and supports tests and insertions of vertices and edges in O(log n) time, where n is the current number of vertices of G. The bounds for tests and vertex insertions are worstcase and the bound for edge insertions is amortized. We also present other applications of this technique to dynamic algorithms for planar graphs.
A LeftFirst Search Algorithm for Planar Graphs
 Discrete Computational Geometry
, 1995
"... We give an O(jV (G)j) time algorithm to assign vertical and horizontal segments to the vertices of any bipartite plane graph G so that (i) no two segments have an interior point in common, (ii) two segments touch each other if and only if the corresponding vertices are adjacent. As a corollary, we o ..."
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Cited by 20 (4 self)
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We give an O(jV (G)j) time algorithm to assign vertical and horizontal segments to the vertices of any bipartite plane graph G so that (i) no two segments have an interior point in common, (ii) two segments touch each other if and only if the corresponding vertices are adjacent. As a corollary, we obtain a strengthening of the following theorem of Ringel and Petrovic. The edges of any maximal bipartite plane graph G with outer face bwb w can be colored by two colors such that the color classes form spanning trees of G b and G b , respectively. Furthermore, such a coloring can be found in linear time. Our method is based on a new linear time algorithm for constructing bipolar orientations of 2connected plane graphs.
Representation of Planar Graphs By Segments
, 1994
"... Given any bipartite planar graph G, one can assign vertical and horizontal segments to its vertices so that (a) no two of them have an interior point in common, (b) two segments have a point in common if and only if the corresponding vertices are adjacent in G. ..."
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Cited by 15 (1 self)
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Given any bipartite planar graph G, one can assign vertical and horizontal segments to its vertices so that (a) no two of them have an interior point in common, (b) two segments have a point in common if and only if the corresponding vertices are adjacent in G.
Optimal reduction of twoterminal directed acyclic graphs
 SIAM Journal on Computing
, 1992
"... Abstract. Algorithms for seriesparallel graphs can be extended to arbitrary twoterminal dags if node reductions are used along with series and parallel reductions. A node reduction contracts a vertex with unit indegree (outdegree) into its sole incoming (outgoing) neighbor. This paper gives an O ..."
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Cited by 14 (1 self)
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Abstract. Algorithms for seriesparallel graphs can be extended to arbitrary twoterminal dags if node reductions are used along with series and parallel reductions. A node reduction contracts a vertex with unit indegree (outdegree) into its sole incoming (outgoing) neighbor. This paper gives an O(n2"5) algorithm for minimizing node reductions, based on vertex cover in a transitive auxiliary graph. Applications include the analysis of PERT networks, dynamic programming approaches to network problems, and network reliability. For NPhard problems one can obtain algorithms that are exponential only in the minimum number of node reductions rather than the number of vertices. This gives improvements if the underlying graph is nearly seriesparallel.
Minimising the Number of Bends and Volume in ThreeDimensional Orthogonal Graph Drawings with a Diagonal Vertex Layout
, 2000
"... A 3D orthogonal drawing of graph with maximum degree at most six positions the vertices at gridpoints in the 3D orthogonal grid, and routes edges along gridlines such that edge routes only intersect at common endvertices. In this paper we present two algorithms for producing 3D orthogonal grap ..."
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Cited by 7 (4 self)
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A 3D orthogonal drawing of graph with maximum degree at most six positions the vertices at gridpoints in the 3D orthogonal grid, and routes edges along gridlines such that edge routes only intersect at common endvertices. In this paper we present two algorithms for producing 3D orthogonal graph drawings with the vertices positioned along the main diagonal of a cube, so called diagonal drawings. This vertexlayout strategy was introduced in the 3Bends algorithm of Eades et al. [11]. We show that minimising the number of bends in a diagonal drawing of a given graph is NPhard. Our first algorithm minimises the total number of bends for a fixed ordering of the vertices along the diagonal. Using two heuristics for determining this vertex ordering we obtain upper bounds on the number of bends. Our second algorithm, which is a variation of the abovementioned 3Bends algorithm, produces 3bend drawings with n^3 + o(n^3) volume, which is the best known upper bound for the volume of 3D orthogonal graph drawings with at most 3 bends per edge.
Balanced VertexOrderings of Graphs
, 2002
"... We consider the problem of determining a balanced ordering of the vertices of a graph; that is, the neighbors of each vertex v are as evenly distributed to the left and right of v as possible. This problem, which has applications in graph drawing for example, is shown to be NPhard, and remains N ..."
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Cited by 4 (3 self)
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We consider the problem of determining a balanced ordering of the vertices of a graph; that is, the neighbors of each vertex v are as evenly distributed to the left and right of v as possible. This problem, which has applications in graph drawing for example, is shown to be NPhard, and remains NPhard for bipartite simple graphs with maximum degree six. We then describe and analyze a number of methods for determining a balanced vertexordering, obtaining optimal orderings for directed acyclic graphs and graphs with maximum degree three. Finally we
Directional Routing via Generals stNumberings
 SIAM JOURNAL ON DISCRETE MATHEMATICS
, 2000
"... We present a mathematical model for network routing based on generating paths in a consistent direction. Our development is based on an algebraic and geometric framework for defining a directional coordinate system for real vector spaces. Our model, which generalizes graph stnumberings, is based on ..."
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Cited by 3 (1 self)
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We present a mathematical model for network routing based on generating paths in a consistent direction. Our development is based on an algebraic and geometric framework for defining a directional coordinate system for real vector spaces. Our model, which generalizes graph stnumberings, is based on mapping the nodes of a network to points in multidimensional space and ensures that the paths generated in di#erent directions from the same source are nodedisjoint. Such directional embeddings encode the global disjoint path structure with very simple local information. We prove that all 3connected graphs have 3directional embeddings in the plane so that each node outside a set of extreme nodes has a neighbor in each of the three directional regions defined in the plane. We conjecture that the result generalizes to kconnected graphs. We also showthat a directed acyclic graph (dag) that is kconnected to a set of sinks has a kdirectional embedding in (k  1)space with the sink set as the extreme nodes.
Finding Double Euler Trails of Planar Graphs in Linear Time
 in Proceedings of the 40th IEEE Symposium on Foundations of Computer Science (FOCS),319329,New York City,1999
, 1999
"... This paper answers an open question in the design of complimentary metaloxide semiconductor VLSI circuits. The question asks whether a polynomialtime algorithm can decide if a given planar graph has a plane embedding has an Euler trail P = e 1 e 2 . . . em and its dual graph has an Euler trai ..."
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Cited by 1 (1 self)
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This paper answers an open question in the design of complimentary metaloxide semiconductor VLSI circuits. The question asks whether a polynomialtime algorithm can decide if a given planar graph has a plane embedding has an Euler trail P = e 1 e 2 . . . em and its dual graph has an Euler trail P # = e # 1 e # 2 . . . e # m , where e # i is the dual edge of e i for i = 1, 2, . . . , m. This paper answers this question in the a#rmative by presenting a lineartime algorithm.