P. Eades, B. D. McKay, and N. C. Wormald, On an edge crossing problem. In Proc. 9th Australian Computer Science Conference, pp. 327-334, Australian National University, 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....classes of graphs. Another approach is to characterize the graphs whose drawings occupy a speci c number of layers and then design ecient algorithms for determining whether or not a given graph satis es the corresponding characterization. So far this approach has been used for only 2 layer [11, 6, 3] and 3 layer [10, 3] drawings. 8 Acknowledgements I thank my supervisor Sue Whitesides for proof reading and helping to improve the readability of this document. This work began as an attempt to reconstruct a proof. I thank David R. Wood for suggesting that reconstruction, and I thank him and ....

Peter Eades, Brendan McKay, and Nick Wormald. On an edge crossing problem. In Proceedings of the 9th Australian Computer Science Conference, pages 327-334. Australian National University, 1986.

....classes of graphs. Another approach is to characterize the graphs whose drawings occupy a specific number of layers and then design e#cient algorithms for determining whether or not a given graph satisfies the corresponding characterization. So far this approach has been used for only 2 layer [11, 6, 3] and 3 layer [10, 3] drawings. 8 Acknowledgements I thank my supervisor Sue Whitesides for proof reading and helping to improve the readability of this document. This work began as an attempt to reconstruct a proof. I thank David R. Wood for suggesting that reconstruction, and I thank him and ....

Peter Eades, Brendan McKay, and Nick Wormald. On an edge crossing problem. In Proceedings of the 9th Australian Computer Science Conference, pages 327--334. Australian National University, 1986. 21

....a graph that admits a biplanar drawing with as the ordering of the vertices in A. The 1 Layer Planarization problem asks if bpr(G; k. Biplanar graphs are easily characterized, and there is a simple linear time algorithm to recognize biplanar graphs, as the next lemma makes clear. Lemma 1 ([4, 8, 17]) Let G be a graph. The following are equivalent: a) G is biplanar. b) G is a forest of caterpillars (see Fig. 2) c) G is acyclic and contains no 2 claw. d) The graph obtained from G by deleting all leaves is a forest and contains no vertex of degree three or greater. Fig. 2. A biplanar ....

P. Eades, B. D. McKay, and N. C. Wormald. On an edge crossing problem. In Proc. 9th Australian Computer Science Conference, pages 327-334. Australian National University, 1986.

....a planar embedding is not possible, then we may want to determine a level embedding with as few crossing arcs as possible. This second crossing minimization problem is known to be NP hard [4] even if there are only two levels with a xed order of nodes on one level as shown by Eades and Wormald [3]; while the rst mentioned planarity test can be performed in linear time by an involved algorithmic approach [11] In this note we present formulations of these problems in terms of CNF formulas, here the planarity test essentially reduces to testing satis ability of a 2CNF formula. The ....

....A and f B , and its feasible solutions must satisfy all clauses of f A and as many clauses of f B as possible. As already mentioned the crossing minimization problem is known to be NP hard [4] even if there are only two levels with a xed order of nodes on one level as shown by Eades and Wormald [3]. Therefore, a lot of e ort (see [2] has been spent to develop ecient heuristics for reducing the number of crossings in drawings of 2 level graphs and to perform a layer by layer sweep for the general k level problem. The most commonly used heuristic, the barycenter heuristic, has been ....

P. Eades, B. McKay and N. C. Wormald, On an edge crossing problem, In Proc. 9th Australian Computer Science Conference, Australian National University (1986), pp. 327-334.

....on each layer i (and not on their absolute position) we can say that the objective is to find the presentation #G, # 0 ,# 1 # of G that minimizes crossings. Gary and Johnson [5] proved that the two layer edge crossing minimization problem is NP hard. More recently, Eades, McKay, and Wormald [2] proved that the problem is NP hard even if the permutation of nodes on one layer is fixed. The problem of determining the minimum cardinality set of edges whose removal allows G to be drawn with no crossings is also NP hard, whether or not the order on one of the layers is fixed [3] This paper ....

P. Eades, B. D. McKay, and N. C. Wormald, On an edge crossing problem,inProc.Ninth Australian Computer Science Conference, 1986, pp. 327--334.

....does not create cycles. Because reversing back edges makes them into forward edges, all cycles are broken by this procedure. It seems reasonable to try to reverse a smaller or even minimal set of edges. One difficulty is that finding a minimal set (the feedback arc set problem) is NP complete [EMW] [GJ] More important, this would probably not improve the drawings. We implemented a heuristic to reverse edges that participate in many cycles. The heuristic takes one non trivial strongly connected component at a time, in an arbitrary order. Within each component, it counts the number of times ....

....multi edges are merged as in the previous pass. The vertex order within ranks determines the edge crossings in the layout, so a good ordering is one with few crossings. Heuristics are appropriate since minimizing edge crossings in layouts of ranked graphs is NP complete, even for only two ranks [EMW]. Several important heuristics for reducing edge crossings in ranked graphs are based on the following scheme first suggested by Warfield [Wa] An initial ordering within each rank is computed. Then a sequence of iterations is performed to try to improve the orderings. Each iteration traverses ....

Eades, P., B. McKay and N. Wormald, "On an Edge Crossing Problem," Proc. 9th Australian Computer Science Conf., 1986, pp. 327-334.

....the partition clearly and at the same time have no crossings. This problem has been studied extensively for bipartite graphs, and for drawings as in the left picture of Figure 1 (so called LL drawings) It is known that a bipartite graph has a planar LL drawing if and only if it is a caterpillar [10, 6]. Exponential time algorithms for minimizing the number of crossings in an LL drawing of a bipartite graph are known as well [12] Two other models of drawing planar partitions, and some surprising applications in boolean functions, were studied Knuth [14] and by Kratochvil and Krivanek [15] ....

....0, while the vertices in B have y coordinate 1. Figure 2: An LL drawing and a graph that has no planar LL drawing. In this, as in all following figures, the vertices of A are drawn white, and the vertices of B are drawn black. Harary and Schwenk [10] and independently Eades, McKay, and Wormald [6] showed that a connected bipartite planar partition has a planar LL drawing if and only if it is a caterpillar, i.e. a graph which is a path after removing all vertices of degree 1. In Section 4, we study planar LL drawings of non bipartite planar partitions, and provide necessary and sufficient ....

P. Eades, B. D. McKay, and N. Wormald. On an edge crossing problem. In ACSC 9, 9th Australian Computer Science Conference, pages 327--334, 1986.

....algorithm have been examined intensively. For phase one we refer to [Sug84] GKNV93] or [EL91] Considering phase two, Garey and Johnson [GJ83] showed that the problem of minimizing the number of edge crossings is NP complete for k level graphs, even if k = 2. According to Eades et al. [EMW86], the problem remains NP complete even if the order of vertices on one of the two levels is fixed. A lot of effort was spent to design efficient heuristics or exact methods such as branch and cut algorithms for crossing reduction in 2 level graphs (see [JM97] or consult [DETT94] for a list of ....

P. Eades, B. D. McKay, and N. C. Wormald. On an edge crossing problem. In Proc. 9th Australian Computer Science Conference, pages 327--334, 1986.

....levels, removing a minimum number of edges such that the resulting subgraph is level planar. For the nal diagram the removed edges are reinserted into a level planar drawing. However, the level planarization problem is NP hard [3] Based on a characterization of 2 level planar graphs (see, e.g. [2]) Mutzel [6] gives an integer linear programming formulation for the 2 level planarization problem, studying the polytope associated with the set of all level planar subgraphs of a level graph with 2 levels. In order to attack the level planarization problem for k 2 levels, an integer linear ....

P. Eades, B. D. McKay, and N. C. Wormald. On an edge crossing problem. In Proc. 9th Australian Computer Science Conference, pages 327-334. Australian National University, 1986.

....clearly and at the same time have no crossings. Different models have been developed of what is meant by displaying the partition ; a unifying naming scheme of those can be found in [1] For the special case of bipartite graphs, this problem has been solved for a number of models (see e.g. [13, 9, 17, 11]) For arbitrary planar partitions, no results are known to the author except the ones in the preceding papers of this series [1, 3] In these papers, we studied three models of drawing planar partitions, and provided necessary and sufficient conditions for the existence of drawing in these ....

P. Eades, B. D. McKay, and N. Wormald. On an edge crossing problem. In ACSC 9, 9th Australian Computer Science Conference, pages 327--334, 1986.

....for P CROSS (G) Other classes of facet defining inequalities can be derived from the following characterization of 2 layer planar graphs (these are graphs that can be drawn on two layers without crossings) using forbidden subgraphs. Theorem 6. Harary and Schwenk, 1972, Tomii et al. 1977, Eades et al. 1986] A 2 layer graph is 2 layer planar if and only if it contains no cycle and no 3 claw. In other words, whenever there is a cycle or a 3 claw (see Figure 1 below) in the graph, we have at least one crossing. Moreover, the exact 2 layer crossing number for cycle graphs is known. 1 5 6 7 2 3 4 1 2 3 ....

Eades, P., McKay, B.D., Wormald, N.C.: On an edge crossing problem. Proc. 9th Australian Computer Science Conference, Australian National University (1986) 327--334.

....algorithm mixes the spring embedder and simulated annealing approaches[FR91] All of the published algorithms for drawing general undirected graphs are heuristic. Indeed, the individual problems of achieving uniform edge lengths and of minimizing the number of edge crossings are both NP hard [EMW86] [GJ83] MO85] Hence, an optimal solution is for all practical purposes unachievable. Rather, people have proposed a variety of heuristics to produce near optimal drawings of graphs. In general, their algorithms have some theoretical foundation, but the only way to evaluate their performance is ....

P. Eades, B. McKay, and N. Wormald. "On an edge crossing problem. " In Proceedings of the Ninth Australian Computer Science Conference, pages 327--334. Australian National University, 1986.

....a layering algorithm which achieves this. 4.1.3 Stage 3: Minimise the number of edge crossings The number of edge crossings depends only on the ordering of the vertices (and dummy vertices) in each layer. Unfortunately, determining this optimum ordering is NP hard [88] even for two layers [89]. Most heuristic algorithms which have been proposed to find approximate solutions fall into two categories. The most common approach is to place the vertices at some form of average of their adjacent vertices positions. The vertex ordering is then implicit in the ordering of coordinate values. ....

P. Eades, B. McKay, and N. Wormald. On an edge crossing problem. In Proc. 9th Australian Computer Science Conference, pages 327--334, 1986.

....it has been shown that all such graphs have an upward drawing. But the resulting drawings do not directly reflect the bipartiteness property in the desired sense. For the full constrained model, necessary and sufficient conditions for the drawability of the graphs are known and can be found in [13, 4]. We will review this model, analyse various other models and give algorithms to decide whether a graph can be drawn without an edge crossing in the specified model or not. Models. Since the class of graphs that can be drawn without edge crossings in the full constrained model is very restricted ....

....connected components separately. 2 The Full Constrained Model Definition 2.1 A vertex will be called small if it has degree one, and large otherwise. Definition 2.2 A connected graph G is called a caterpillar if G is a tree and each vertex in G has at most two large neighbours. Theorem 2. 1 [13, 4] A connected graph G is in B fc if and only if G is a caterpillar. A corresponding drawing can be found in linear time. The proof is simple. It consists of a straightforward algorithm and an easy contradiction argument. Details are omitted [13, 4] The full constrained model allows a planar ....

[Article contains additional citation context not shown here]

Eades P., B.D. McKay and N. Wormald, `On an edge crossing problem', Proc. Australian Computer Science Conf., Austr. Nat. Univ. (1986), pp. 327--334.

....where rank is used to determine the vertical placement of nodes. Dummy nodes are often added so that no edge crosses several levels. 2. Order the nodes from left to right within a rank to reduce the number of edge crossings in the layout. Minimizing the number of edge crossings is NP Complete [PEW86]. A heuristic to avoid this complexity follows. Given an initial ordering within a rank, a sequence of iterations is performed to improve the orderings. An iteration traverses through the ranks from first to last or vice versa. Each vertex within a rank is assigned a weight based on the relative ....

....be further classified. A cross edge connects nodes that are unrelated in the partial order. A forward edge connects a node to a descendant. A back edge connects a descendant to an ancestor. A cycle can be broken by reversing a back edge. Determining a minimum set of edges to reverse is NPcomplete [PEW86]. The heuristic used takes each non trivial strongly connected component, counts the number of times each edge forms a cycle, and reverses the edge with the maximum count. The initial step in ranking the acyclic graph is to determine a feasible spanning tree and a cut value for each edge. The cut ....

B. McKay P. Eades and N. Wormald. On an edge crossing problem. In Proceedings 9th Australian Computer Science Conference, pages 175--198, 1986.

....planar graphs in terms of subgraphs. We will call the graph shown in Figure 3(a) a double claw. A caterpillar is a connected graph G = V; E) having edges on its backbone (v 1 ; v 2 ; v l ) and single edges (v i ; w) w 2 V n fv 1 ; v 2 ; v l g (see Figure 3(b) Theorem 1 [15, 3]. A two layer graph is two layer planar if and only if it contains no cycle and no double claw. a) b) c) v 1 v 3 v 2 v 4 v 6 v 5 v 7 v 8 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 Figure 3: a) Double claw. b) Caterpillar. c) Caterpillars can be embedded on two layers without any crossings. ....

P. Eades, B. D. McKay, and N. C. Wormald. On an edge crossing problem. In Proc. 9th Australian Computer Science Conference, pages 327--334, Australian National University, 1986.

....in many areas, such as software engineering [23] knowledge representation [13] idea organization [14] software visualization [22] and VLSI design [10] Planarity is a much studied area for classical graphs. For example, the problem of minimizing edge crossings is proved to be NP hard [9, 7]. However, efficient algorithms for testing whether a graph is planar (i.e. can be drawn without edge crossings) exist [11, 16, 3, 6] Planarity issues relating to the more powerful graph models mentioned above have not been studied. In this paper, we introduce C planarity, the planarity of ....

P. Eades, B. McKay, and N. Wormald. On an edge crossing problem. In Proc. 9th Australian Computer Science Conf., pages 327--334, 1986.

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P. Eades, B. D. McKay, and N. C. Wormald, On an edge crossing problem. In Proc. 9th Australian Computer Science Conference, pp. 327-334, Australian National University, 1986.

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