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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 9 (6 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.
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.
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
GRAPH SEARCHING AND RELATED PROBLEMS
"... Abstract. Suppose that there is a robber hiding on vertices or along edges of a graph or digraph. Graph searching is concerned with finding the minimum number of searchers required to capture the robber. We survey the major results of graph searching problems, focusing on algorithmic, structural, a ..."
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Abstract. Suppose that there is a robber hiding on vertices or along edges of a graph or digraph. Graph searching is concerned with finding the minimum number of searchers required to capture the robber. We survey the major results of graph searching problems, focusing on algorithmic, structural, and probabilistic aspects of the field. 1.
Degreeconstrained orientations of embedded graphs
"... Abstract. We investigate the problem of orienting the edges of an embedded graph in such a way that the indegrees of both the nodes and faces meet given values. We show that the number of feasible solutions is bounded by 22g, where g is the genus of the embedding, and all solutions can be determi ..."
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Abstract. We investigate the problem of orienting the edges of an embedded graph in such a way that the indegrees of both the nodes and faces meet given values. We show that the number of feasible solutions is bounded by 22g, where g is the genus of the embedding, and all solutions can be determined within time O(22gE2 + E3). In particular, for planar graphs the solution is unique if it exists, and in general the problem of finding a feasible orientation is fixedparameter tractable in g. In sharp contrast to these results, we show that the problem becomes NPcomplete even for a fixed genus if only upper and lower bounds on the indegrees are specified instead of exact values. 1
Bounded Degree Acyclic Decompositions of Digraphs
, 2002
"... An acyclic decomposition of a digraph is a partition of the edges into acyclic subgraphs. Trivially every digraph has an acyclic decomposition into two subgraphs. We prove that for every integer s 2 every digraph has an acyclic decomposition into s subgraphs such that in each subgraph the outdegree ..."
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An acyclic decomposition of a digraph is a partition of the edges into acyclic subgraphs. Trivially every digraph has an acyclic decomposition into two subgraphs. We prove that for every integer s 2 every digraph has an acyclic decomposition into s subgraphs such that in each subgraph the outdegree of each vertex v is at most d s 1 e. For all digraphs this degree bound is optimal.