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25
How to morph tilings injectively
 Journal of Computational and Applied Mathematics
, 1999
"... Abstract: We describe a method based on convex combinations for morphing corresponding pairs of tilings in IR 2. It is shown that the method always yields a valid morph when the boundary polygons are identical, unlike the standard linear morph. Key words: Morphing, triangulations, tilings, polygons, ..."
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Cited by 34 (13 self)
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Abstract: We describe a method based on convex combinations for morphing corresponding pairs of tilings in IR 2. It is shown that the method always yields a valid morph when the boundary polygons are identical, unlike the standard linear morph. Key words: Morphing, triangulations, tilings, polygons, convex combinations. 1.
Convex drawings of Planar Graphs and the Order Dimension of 3Polytopes
 ORDER
, 2000
"... We define an analogue of Schnyder's tree decompositions for 3connected planar graphs. Based on this structure we obtain: Let G be a 3connected planar graph with f faces, then G has a convex drawing with its vertices embedded on the (f 1) (f 1) grid. Let G be a 3connected planar graph. ..."
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Cited by 32 (12 self)
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We define an analogue of Schnyder's tree decompositions for 3connected planar graphs. Based on this structure we obtain: Let G be a 3connected planar graph with f faces, then G has a convex drawing with its vertices embedded on the (f 1) (f 1) grid. Let G be a 3connected planar graph. The dimension of the incidence order of vertices, edges and bounded faces of G is at most 3. The second result is originally due to Brightwell and Trotter. Here we give a substantially simpler proof.
Planar Minimally Rigid Graphs and PseudoTriangulations
, 2003
"... Pointed pseudotriangulations are planar minimally rigid graphs embedded in the plane with pointed vertices (incident to an angle larger than π). In this paper we prove that the opposite statement is also true, namely that planar minimally rigid graphs always admit pointed embeddings, even under cer ..."
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Cited by 30 (14 self)
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Pointed pseudotriangulations are planar minimally rigid graphs embedded in the plane with pointed vertices (incident to an angle larger than π). In this paper we prove that the opposite statement is also true, namely that planar minimally rigid graphs always admit pointed embeddings, even under certain natural topological and combinatorial constraints. The proofs yield efficient embedding algorithms. They also provide—to the best of our knowledge—the first algorithmically effective result on graph embeddings with oriented matroid constraints other than convexity of faces.
Spring Algorithms and Symmetry
 Theoretical Computer Science
, 1999
"... Spring algorithms are regarded as effective tools for visualizing undirected graphs. One major feature of applying spring algorithms is to display symmetric properties of graphs. This feature has been confirmed by numerous experiments. In this paper, firstly we formalize the concepts of graph symmet ..."
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Cited by 23 (3 self)
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Spring algorithms are regarded as effective tools for visualizing undirected graphs. One major feature of applying spring algorithms is to display symmetric properties of graphs. This feature has been confirmed by numerous experiments. In this paper, firstly we formalize the concepts of graph symmetries in terms of "reflectional" and "rotational" automorphisms; and characterize the types of symmetries, which can be displayed simultaneously by a graph layout, in terms of "geometric" automorphism groups. We show that our formalization is complete. Secondly, we provide general theoretical evidence of why many spring algorithms can display graph symmetry. Finally, the strength of our general theorem is demonstrated from its application to several existing spring algorithms. 1 Introduction Graphs are commonly used in Computer Science to model relational structures such as programs, databases, and data structures. A good graph "layout" gives a clear understanding of a structural model; a ba...
Strictly Convex Drawings of Planar Graphs
, 2004
"... Every threeconnected planar graph with n vertices has a drawing on an O(n7=3) \Theta O(n7=3) grid in which all faces are strictly convex polygons. ..."
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Cited by 19 (1 self)
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Every threeconnected planar graph with n vertices has a drawing on an O(n7=3) \Theta O(n7=3) grid in which all faces are strictly convex polygons.
An Algorithm For Drawing A Hierarchical Graph
, 1995
"... this paper we present a method for drawing "hierarchical directed graphs", which are digraphs in which each node is assigned a layer, as in Figure 1. ..."
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Cited by 18 (7 self)
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this paper we present a method for drawing "hierarchical directed graphs", which are digraphs in which each node is assigned a layer, as in Figure 1.
Upward Planar Drawing of Single Source Acyclic Digraphs
, 1990
"... A upward plane drawing of a directed acyclic graph is a straight line drawing in the Euclidean plane such that all directed arcs point upwards. Thomassen [30] has given a nonalgorithmic, graphtheoretic characterization of those directed graphs with a single source that admit an upward drawing. We ..."
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Cited by 15 (0 self)
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A upward plane drawing of a directed acyclic graph is a straight line drawing in the Euclidean plane such that all directed arcs point upwards. Thomassen [30] has given a nonalgorithmic, graphtheoretic characterization of those directed graphs with a single source that admit an upward drawing. We present an efficient algorithm to test whether a given singlesource acyclic digraph has a plane upward drawing and, if so, to find a representation of one such drawing. The algorithm decomposes the graph into biconnected and triconnected components, and defines conditions for merging the components into an upward drawing of the original graph. For the triconnected components we provide a linear algorithm to test whether a given plane representation admits an upward drawing with the same faces and outer face, which also gives a simpler (and algorithmic) proof of Thomassen's result. The entire testing algorithm (for general single source directed acyclic graphs) operates in O(n²) time and...
Heuristics and Experimental Design for Bigraph Crossing Number Minimization
 IN ALGORITHM ENGINEERING AND EXPERIMENTATION (ALENEX’99), NUMBER 1619 IN LECTURE NOTES IN COMPUTER SCIENCE
, 1999
"... The bigraph crossing problem, embedding the two vertex sets of a bipartite graph G = (V0;V1;E) along two parallel lines so that edge crossings are minimized, has application to circuit layout and graph drawing. We consider the case where both V0 and V1 can be permuted arbitrarily  both this and ..."
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Cited by 14 (9 self)
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The bigraph crossing problem, embedding the two vertex sets of a bipartite graph G = (V0;V1;E) along two parallel lines so that edge crossings are minimized, has application to circuit layout and graph drawing. We consider the case where both V0 and V1 can be permuted arbitrarily  both this and the case where the order of one vertex set is fixed are NPhard. Two new heuristics that perform well on sparse graphs such as occur in circuit layout problems are presented. The new heuristics outperform existing heuristics on graph classes that range from applicationspecific to random. Our experimental design methodology ensures that differences in performance are statistically significant and not the result of minor variations in graph structure or input order.
Symmetric Drawings of Triconnected Planar Graphs
 Proc. of SODA 2002
, 2002
"... Abstract: This paper proves that every internally triconnected hierarchical plane graph with the outer facial cycle drawn as a convex polygon admits a convex drawing. We present an algorithm which constructs such a drawing. This extends the previous known result that every hierarchical plane graph a ..."
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Cited by 9 (7 self)
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Abstract: This paper proves that every internally triconnected hierarchical plane graph with the outer facial cycle drawn as a convex polygon admits a convex drawing. We present an algorithm which constructs such a drawing. This extends the previous known result that every hierarchical plane graph admits a straightline drawing.
Recent Excluded Minor Theorems for Graphs
 IN SURVEYS IN COMBINATORICS, 1999 267 201222. THE ELECTRONIC JOURNAL OF COMBINATORICS 8 (2001), #R34 8
, 1999
"... A graph is a minor of another if the first can be obtained from a subgraph of the second by contracting edges. An excluded minor theorem describes the structure of graphs with no minor isomorphic to a prescribed set of graphs. Splitter theorems are tools for proving excluded minor theorems. We disc ..."
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Cited by 9 (0 self)
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A graph is a minor of another if the first can be obtained from a subgraph of the second by contracting edges. An excluded minor theorem describes the structure of graphs with no minor isomorphic to a prescribed set of graphs. Splitter theorems are tools for proving excluded minor theorems. We discuss splitter theorems for internally 4connected graphs and for cyclically 5connected cubic graphs, the graph minor theorem of Robertson and Seymour, linkless embeddings of graphs in 3space, Hadwiger’s conjecture on tcolorability of graphs with no Kt+1 minor, Tutte’s edge 3coloring conjecture on edge 3colorability of 2connected cubic graphs with no Petersen minor, and Pfaffian orientations of bipartite graphs. The latter are related to the even directed circuit problem, a problem of Pólya about permanents, the 2colorability of hypergraphs, and signnonsingular matrices.