Feng, Q., Algorithms for Drawing Clustered Graphs, Ph.D. thesis, Department of Computer Science and Software Engineering, University of Newcastle, Apr. 1997.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....with maximal edge connectivity. We are going to prove, that communities are unique and completely determined by the communities of vertices and edges. Furthermore we are going to see that the communities in a graph G can be partially ordered, resulting in a natural clustering of G (cmp. FCE95] Fen97] More detailled, we obtain a tree of subgraphs whose leaves are the vertices of G, whose inner node represent the communities. Furthermore the children of a node are disjoint and completely contained in the parent. In section 4 we are going to extend our de nition of communities to communities ....

....the communities of all connected subgraphs. In this section we are going to describe how they can be calculated and represented quite eciently. For the representation of all vertex and edge communities of a graph G = V; E) we use clustered graphs as introduced by Feng et al. in [FCE95] and [Fen97] The foundation is a directed tree T which has precisely kV k leaves together with a bijection : leaves(T ) V , associating a unique vertex of G to each leaf of T . We assume that all edges of T are directed from parent to child. The pair (T ; is called a clustering of G. We inductively ....

Qingwen Feng. Algorithms for Drawing Clustered Graphs. PhD thesis, Department of Computer Science and Software Engineering, University of Newcastle, 1997.

....levels. A natural realization of such multiple level representations is a 3D drawing with each level drawn on a plane at a di erent z coordinate, and with the clustering structure drawn as a tree in 3D. 2 The multi level display algorithms are introduced by Eades and Feng [13] and Feng [16] in the context of visualization for clustered graphs. Compound and clustered graphs are studied by Sugiyama and Misue [30, 37] by North [32] by Eades, Feng and Lin [14] and Feng, Cohen, and Eades [17] Creating a graph clustering based on binary space partitions and using it to display large ....

Q. Feng. Algorithms for Drawing Clustered Graphs. PhD thesis, Department of Computer Science and Software Engineering, University of Newcastle, Apr. 1997.

....graph is c planar if there are no edge crossings or edge region crossings. If a clustered graph C has a c planar drawing then we say that it is c planar. A clustered graph C = G, T ) is a connected clustered graph if each cluster induces a connected subgraph of G. The following results from [14, 16] characterize c planarity in a way which can be exploited by our drawing algorithms. Theorem 1 A connected clustered graph C = G, T ) is c planar if and only if the graph G is planar and there exists a planar drawing D of G, such that for each node # of T , all the vertices and edges of G G(#) ....

....straight line drawing of a clustered graph C = G, T ) edges are required to be drawn as straight lines and clusters must be drawn as convex polygons. The question of whether every c planar clustered graph admits a planar straightline drawing or not has been studied and answered a#rmatively [9, 10, 14]. However, it is still open whether or not more regular convex bodies such as circles and rectangles can be used for clusters in a planar straight line drawing. The results of this paper together with those in [9, 10] imply that a c planar clustered graph C = G, T ) admits a planar straight line ....

Q. Feng. Algorithms for Drawing Clustered Graphs. PhD thesis, Department of Computer Science and Software Engineering, University of Newcastle, 1997.

....are an important class [2] of graphs to be investigated in this area. The upward drawing convention for drawing acyclic directed graphs has received a great deal of attention since last decade; and a number of results for drawing upward planar graphs have been published [2, 4, 6, 11] 1 Consider [7, 8, 9, 19] that directed graphs are not powerful enough to model every real life application. Hierarchical graphs are then introduced, where layering information is added to a directed acyclic graph. Consequently, the hierarchical drawing convention is proposed to display the speci ed layering ....

Q. Feng, Algorithms for Drawing Clustered Graphs, PhD thesis, Department of Computer Science and Software Engineering, University of Newcastle, 1997.

....abstraction levels. A natural realization of such multiple level representations is a 3D drawing with each level drawn on a plane at a di#erent z coordinate, and with the clustering structure drawn as a tree in 3D. The multi level display algorithms are introduced by Eades and Feng [13] and Feng [16] in the context of visualization for clustered graphs. Compound and clustered graphs are studied by Sugiyama and Misue [30, 37] by North [32] by Eades, Feng and Lin [14] and Feng, Cohen, and Eades [17] Creating a graph clustering based on binary space partitions and using it to display large ....

Q. Feng. Algorithms for Drawing Clustered Graphs. PhD thesis, Department of Computer Science and Software Engineering, University of Newcastle, Apr. 1997.

....n ) Figure 24 shows a drawing of C 4 . Future work on hierarchical graphs and clustered graphs should address the following open problems: ffl We note that relaxing the straight line constraints can give us polynomial area bounds both in hierarchical drawings and in convex cluster drawings [18, 12, 13]. In future work, we would like to investigate the trade off between the number of bends and the area of the drawing. ffl The drawings of clustered graphs may lack vertical compaction because we use an st numbering as the layer assignment. It is very worthwhile to investigate methods that can ....

Q. Feng. Algorithms for Drawing Clustered Graphs. PhD thesis, Department of Computer Science and Software Engineering, University of Newcastle, 1997.

....output by our algorithm. are no edge crossings or edge region crossings. If a clustered graph C has a c planar drawing then we say that it is c planar. A clustered graph C = G; T ) is a connected clustered graph if each cluster induces a connected subgraph of G. The following results from [15, 17] characterize c planarity in a way which can be exploited by our drawing algorithms. Theorem 1 A connected clustered graph C = G; T ) is c planar if and only if the graph G is planar and there exists a planar drawing D of G, such that for each node of T , all the vertices and edges of G G( ....

....are required to be drawn as straight lines and clusters must be drawn as convex polygons. The question Figure 16: Simulation of a degree 10 vertex with a cluster. of whether every c planar clustered graph admits a planar straight line drawing or not has been studied and answered a rmatively [10, 11, 15]. However, it is still open whether or not more regular convex bodies such as circles and rectangles can be used for clusters in a planar straight line drawing. The results of this paper together with those in [10, 11] imply that a c planar clustered graph C = G; T ) admits a planar straight line ....

Q. Feng. Algorithms for Drawing Clustered Graphs. PhD thesis, Department of Computer Science and Software Engineering, University of Newcastle, 1997.

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Feng, Q., Algorithms for Drawing Clustered Graphs, Ph.D. thesis, Department of Computer Science and Software Engineering, University of Newcastle, Apr. 1997.

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