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34
On the orthogonal drawing of outerplanar graphs
 in Proceedings of COCOON ’04, Lect. Notes Comput. Sci. 3106
, 2004
"... Abstract. In this paper we show that an outerplanar graph G with maximum degree at most 3 has a 2D orthogonal drawing with no bends if and only if G contains no triangles. We also show that an outerplanar graph G with maximum degree at most 6 has a 3D orthogonal drawing with no bends if and only i ..."
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Abstract. In this paper we show that an outerplanar graph G with maximum degree at most 3 has a 2D orthogonal drawing with no bends if and only if G contains no triangles. We also show that an outerplanar graph G with maximum degree at most 6 has a 3D orthogonal drawing with no bends if and only if G contains no triangles. 1
Orthogonal drawings with few layers
 PROC. 9TH INTERNATIONAL SYMP. ON GRAPH DRAWING (GD '01
, 2002
"... In this paper, we study 3dimensional orthogonal graph drawings. Motivated by the fact that only a limited number of layers is possible in VLSI technology, and also noting that a small number of layers is easier to parse for humans, we study drawings where one dimension is restricted to be very smal ..."
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Cited by 4 (3 self)
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In this paper, we study 3dimensional orthogonal graph drawings. Motivated by the fact that only a limited number of layers is possible in VLSI technology, and also noting that a small number of layers is easier to parse for humans, we study drawings where one dimension is restricted to be very small. We give algorithms to obtain pointdrawings with 3layers and 4 bends per edge, and algorithms to obtain boxdrawings with 2 layers and 2 bends per edge. Several other related results are included as well. Our constructions have optimal volume, which we prove by providing lower bounds.
Lower Bounds for the Number of Bends in ThreeDimensional Orthogonal Graph Drawings
, 2003
"... This paper presents the first nontrivial lower bounds for the total number of bends in 3D orthogonal graph drawings with vertices represented by points. In particular, we prove lower bounds for the number of bends in 3D orthogonal drawings of complete simple graphs and multigraphs, which are tigh ..."
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This paper presents the first nontrivial lower bounds for the total number of bends in 3D orthogonal graph drawings with vertices represented by points. In particular, we prove lower bounds for the number of bends in 3D orthogonal drawings of complete simple graphs and multigraphs, which are tight in most cases. These result are used as the basis for the construction of infinite classes of cconnected simple graphs, multigraphs, and pseudographs (2 &le; c &le; 6) of maximum degree &Delta; (3 &le; &Delta; &le; 6), with lower bounds on the total number of bends for all members of the class. We also present lower bounds for the number of bends in general position 3D orthogonal graph drawings. These results have significant ramifications for the `2bends problem', which is one of the most important open problems in the field.
The Topology of Bendless ThreeDimensional Orthogonal Graph Drawing
, 2007
"... We consider embeddings of 3regular graphs into 3dimensional Cartesian coordinates, in such a way that two vertices are adjacent if and only if two of their three coordinates are equal (that is, if they lie on an axisparallel line) and such that no three points lie on the same axisparallel line; ..."
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Cited by 2 (1 self)
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We consider embeddings of 3regular graphs into 3dimensional Cartesian coordinates, in such a way that two vertices are adjacent if and only if two of their three coordinates are equal (that is, if they lie on an axisparallel line) and such that no three points lie on the same axisparallel line; we call a graph with such an embedding an xyz graph. We describe a correspondence between xyz graphs and facecolored embeddings of the graph onto twodimensional manifolds, and we relate bipartiteness of the xyz graph to orientability of the underlying topological surface. Using this correspondence, we show that planar graphs are xyz graphs if and only if they are bipartite, cubic, and threeconnected, and that it is NPcomplete to determine whether an arbitrary graph is an xyz graph. We also describe an algorithm with running time O(n2 n/2) for testing whether a given graph is an xyz graph.
private communication
, 2005
"... We show that every graph of maximum degree three can be drawn without crossings in three dimensions with at most two bends per edge, and with 120 ◦ angles between all pairs of edge segments that meet at a vertex or a bend. We show that every graph of maximum degree four can be drawn in three dimensi ..."
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We show that every graph of maximum degree three can be drawn without crossings in three dimensions with at most two bends per edge, and with 120 ◦ angles between all pairs of edge segments that meet at a vertex or a bend. We show that every graph of maximum degree four can be drawn in three dimensions with at most three bends per edge, and with 109.5 ◦ angles, i. e., the angular resolution of the diamond lattice, between all pairs of edge segments that meet at a vertex or a bend. The angles in these drawings are the best possible given the degrees of the vertices. Submitted:
Imbalance is Fixed Parameter Tractable
"... Abstract. In the Imbalance Minimization problem we are given a graph G = (V, E) and an integer b and asked whether there is an ordering v1... vn of V such that the sum of the imbalance of all the vertices is at most b. The imbalance of a vertex vi is the absolute value of the difference between the ..."
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Abstract. In the Imbalance Minimization problem we are given a graph G = (V, E) and an integer b and asked whether there is an ordering v1... vn of V such that the sum of the imbalance of all the vertices is at most b. The imbalance of a vertex vi is the absolute value of the difference between the number of neighbors to the left and right of vi. The problem is also known as the Balanced Vertex Ordering problem and it finds many applications in graph drawing. We show that this problem is fixed parameter tractable and provide an algorithm that runs in time 2 O(b log b) · n O(1). This resolves an open problem of Kára et al. [COCOON 2005]. 1
Bounds for Orthogonal 3D . . .
 J. GRAPH ALGORITHMS APPL
, 1999
"... This paper studies 3D orthogonal grid drawings for graphs of arbitrary degree, in particular Kn , with vertices drawn as boxes. It establishes asymptotic lower bounds for the volume of the bounding box and the number of bends of such drawings and exhibits a construction that achieves these bounds. ..."
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This paper studies 3D orthogonal grid drawings for graphs of arbitrary degree, in particular Kn , with vertices drawn as boxes. It establishes asymptotic lower bounds for the volume of the bounding box and the number of bends of such drawings and exhibits a construction that achieves these bounds. No edge route in this construction bends more than three times. For drawings constrained to have at most k bends on any edge route, simple constructions are given for k =1andk =2.The unconstrained construction handles the k # 3cases.