G.D. Battista and R. Tamassia. Incremental planarity testing (extended abstract). In 30th FOCS, pages 436-441, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of the rst edge on the path to the root, from which we immediately get a parent pointer. Unfortunately, the above axiomatic interface has been found too limited for many application of dynamic trees, and instead authors have worked directly with the Sleator and Tarjan s underlying representation [30, 5, 21, 24, 23, 14, 4, 1, 16, 9, 8, 7, 22]. In particular, this is the case for the previous solutions to the dynamic center [6] and median problems [3] and we believe part of the reason for their worse bounds and more complex solutions is diculties in working directly with Sleator and Tarjan s underlying representation. Of course, one ....

G.D. Battista and R. Tamassia. Incremental planarity testing (extended abstract). In 30th FOCS, pages 436-441, 1989.

....is to incrementalize our variant of Sugiyama s hierarchical drawing algorithm. Some similar issues, though, are encountered in making other kinds of layouts incrementally. 2 Previous Work Significant progress has been made in drawing dynamic trees [15] using subtree contours) planar graphs [2] and series parallel graphs [7] Although these are useful techniques for these restricted classes of graphs, they are not directly applicable to general graph drawing. Hornick, Miriyala and Tamassia describe a practical incremental edge router for orthogonal drawings, such as entity relation ....

G. Di Battista and R. Tamassia. Incremental planarity testing. In Proc. 30th IEEE Symp. on Foundations of Computer Science, pages 436--441, 1989.

....found spanning planar subgraph. The latter algorithm is shown to be incorrect by [Kan92, Kan93] claiming to show how to correct the algorithm. Lei94, JLM97] in turn point out that the result in [Kan92] is not correct either and discuss the di#culties of using PQ trees. Di Battista and Tamassia [DT89, DT96b] DT90, DT96a] define and use SPQR trees to describe the recursive decomposition of a 2 connected graph into its 3 connected components. DT89] obtains an O(m log n) time algorithm for finding a maximal planar subgraph as a byproduct of an algorithm for incremental planarity testing . An ....

....JLM97] in turn point out that the result in [Kan92] is not correct either and discuss the di#culties of using PQ trees. Di Battista and Tamassia [DT89, DT96b] DT90, DT96a] define and use SPQR trees to describe the recursive decomposition of a 2 connected graph into its 3 connected components. DT89] obtains an O(m log n) time algorithm for finding a maximal planar subgraph as a byproduct of an algorithm for incremental planarity testing . An incremental (or dynamic) planarity testing algorithm maintains a data structure representing a planar graph G = V, E) and can handle requests of the ....

Giuseppe Di Battista and Roberto Tamassia. Incremental Planarity Testing. In Proceedings of the 30th Annual Symposium on Foundations of Computer Science, FOCS'89, pages 436--441, 1989.

....decade there has been a growing interest in dynamic problems on graphs. In particular, much attention has been devoted to the dynamic maintenance of connected components [14, 16, 43] and higher connectivity [8, 10, 18, 19, 20, 21, 32, 35, 53] transitive closure [29, 30, 31, 37, 47, 55] planarity [7, 8, 46], shortest paths [2, 5, 13, 39, 44] and minimum spanning trees [10, 11, 16] In these problems one would like to answer queries on graphs that are undergoing a sequence of updates, such as insertions and deletions of edges and vertices. The goal of a dynamic graph algorithm is to update ....

....if only one type of update, i.e. either insertions or deletions, is allowed. Partially dynamic problems that deal with insertions only are called incremental. Planarity testing is a basic problem which has inspired an extensive amount of research in graph theory [9, 34, 54] data structures [4, 7, 15, 46], and sequential [28, 38] as well as parallel [33, 42] algorithms. Informally, a graph is planar if it can be embedded onto the plane without edge crossings. The planarity testing problem consists of answering the question whether a given graph is planar and if so of constructing such an embedding ....

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G. Di Battista and R. Tamassia. Incremental planarity testing. In Proc. 30th Annual Symp. on Foundations of Computer Science, pages 436--441, 1989.

....above described, such approach can be implemented in O( V E ) time complexity. An incremental planarity testing algorithm, based on an O(log V ) time per operation strategy for the problem of maintaining a planar graph under edge additions, was proposed by Di Battista and Tamassia [7]. Hence, their algorithm leads to a more date: November 4, 1998 file: ribeir1 e#cient implementation of the incremental approach for finding a maximal planar subgraph with O( E log V ) time complexity. Two phase heuristics. The heuristics described in this section are based on the ....

Di Battista, G., and Tamassia, R.: `Incremental planarity testing': Proceedings of the 30th IEEE Symp. FOCS, 1989, pp. 436--441.

....The Newbery algorithm does not address how to adjust layouts smoothly when nodes change ranks, nor does it concern geometric stability between frames. Significant progress has been made in the algorithms community on dynamic trees [Moe90] using subtree contours for updates) and planar graphs [BT89] These may be useful buildingblocks for future work, but the results are limited by the restriction to certain kinds of graphs. 3 Incremental layouts Revisiting the basic problem of presenting dynamic graphs, the input is a sequence of graphs G 0 ; G 1 ; G 2 ; G n that we interpret as ....

G. Di Battista and R. Tamassia. Incremental planarity testing. In Proc. 30th IEEE Symp. on Foundations of Computer Science, pages 436--441, 1989.

....for Gm is not empty, the graph G = Gm is obviously leveled planar. As long as the graph G j is connected for some j 2 f1; 2; 3; mg standard PQ tree techniques similar to the ones used in the planarity test can be applied in order to determine the required set of permutations (see Di Battista and Tamassia, 1989). In case that G j , 1 j m, consists of more than one connected component, Heath and Pemmaraju suggest to use a PQ tree for every component and formulate a set of rules of how to merge components F 1 and F 2 , respectively their corresponding PQ trees T 1 and T 2 , if F 1 and F 2 both are ....

Di Battista, G. and Tamassia, R. (1989). Incremental planarity testing. In Proceedings on the 30th Annual IEEE Symposium on Foundations of Computer Science, North Carolina, pages 436--441.

....given that planarity is tested in time O(n) More eOEcient algorithms have been presented in the literature. Cai, Han, and Tarjan [CHT93] uses the path addition algorithm of [HT74] to nd a maximal planar subgraph, achieving an O(m log n) time bound. The algorithm of Di Battista and Tamassia [DT89] checks in O(log n) time whether or not an edge can be added to G without destroying planarity, rendering an O(m log n) algorithm as well. An algorithm claiming to give results closer to optimum than what [DT89] Kan92] see below) and earlier versions of the algorithm of [CHT93] give, has been ....

....achieving an O(m log n) time bound. The algorithm of Di Battista and Tamassia [DT89] checks in O(log n) time whether or not an edge can be added to G without destroying planarity, rendering an O(m log n) algorithm as well. An algorithm claiming to give results closer to optimum than what [DT89], Kan92] see below) and earlier versions of the algorithm of [CHT93] give, has been presented by Goldschmidt and Takvorian [GT94] They use a very interesting approach, but the worst case time complexity of this algorithm is at least O(nm) The rst approach using PQ trees to planarize a non ....

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G. Di Battista and R. Tamassia. Incremental planarity testing. In Proc. 30th Annual Symposium an Foundations of Computer Science, pages 436441. IEEE Comput. Soc. Press, 1989.

....solving this problem. Chiba, Nishioka, and Shirakawa (1979) presented an algorithm based on the path addition algorithm that computes a maximal planar subgraph in O(nm) time. Cai, Han, and Tarjan (1993) presented later an O(m log n) algorithm that is based on the path addition algorithm as well. Di Battista and Tamassia (1989) described an algorithm that checks in O(log n) amortized time, whether an edge can be added to G without destroying planarity, obtaining an O(m log n) time algorithm as well. Ozawa and Takahashi (1981) have presented an O(nm) algorithm using the vertex addition algorithm. Jayakumar, ....

Di Battista, G. and Tamassia, R. (1989). Incremental planarity testing. In Proceedings on the 30th Annual IEEE Symposium on Foundations of Computer Science, North Carolina, pages 436--441.

....as a function of the output change [17, 40] The main dynamic problems considered on directed graphs include shortest paths and transitive closure. For lack of space, we do not include in this chapter dynamic algorithms for planar graphs, which have received considerable attention in recent years [6, 7, 11, 12, 18, 21, 22, 28, 32, 34, 42, 46, 47, 48, 51], and focus our attention to general undirected graphs only. The remainder of the chapter is organized as follows. In Section 2 we give some preliminary definitions and a little terminology. Dynamic tree problems are considered in Section 3, while in Section 4 we describe partially dynamic ....

G. Di Battista and R. Tamassia. Incremental planarity testing. In Proc. 30th IEEE Symp. Foundations of Computer Science, pages 436--441, 1989.

....bonds into maximal sets of multiple edges (bonds) and the triangles into maximal simple cycles (polygons) The triconnected components are unique. Next we need the SPQR tree, a versatile data structure that represents the decomposition of a biconnected graph into its triconnected components [10]. The SPQR tree T is defined as follows: for every triconnected component we create an R node, for every polygon an S node, for every bond a P node, and for every edge a Q node. The edges in T are defined as follows: Let u; v be nodes in T . If u is a Q node then there is an edge between u and v ....

Di Battista, G., and R. Tamassia, Incremental planarity testing, in: Proc. 30th Annual IEEE Symp. on Found. of Comp. Science, North Carolina, 1989, pp. 436--441.

....Future Work 139 ffl In many applications, the relational structure to be visualized is dynamic , and it is inefficient to apply a static drawing algorithm whenever an incremental change is made. Research has been done on developing dynamic drawing algorithms for classical graphs [CBT 92, BT89, ELMS91, PT96, PST97] Dynamic drawing algorithms for clustered graphs have yet to be investigated. 140 ....

G. di Battista and R. Tamassia. Incremental planarity testing. In Proc. 30th IEEE Symp. on Foundations of Computer Science, pages 436--441, 1989.

....and more generalized version of the latter. Cai, Han, and Tarjan [4] proposed an O(jEj log jV j) algorithm for finding a maximal planar subgraph based on the Hopcroft and Tarjan planarity testing algorithm [18] Another O(jEj log jV j) algorithm was also proposed by Di Battista and Tamassia [9], based on an O(log jV j) time per operation strategy to the problem of maintaining a planar graph under edge additions. Cimikowski [5] proposed a heuristic based on finding, for each biconnected component of a nonplanar graph, a pair of edge disjoint spanning trees whose union is October 13, ....

G. Di Battista and R. Tamassia, Incremental planarity testing, in Proceedings of the 30th IEEE Symp. FOCS, Chapel Hill, NC, 1989, pp. 436--441.

....subgraph problem is to determine a planar subgraph H of G such that no edge of G H can be added to H without destroying planarity. Polynomial algorithms have been obtained by Jakayumar, Thulasiraman and Swamy [6] and Wu [9] O(mlogn) algorithms were previously given by Di Battista and Tamassia [3] and Cai, Han and Tarjan [2] A recent O(m a a a a a a a a a a a a a a a a a a a a a a a a a (n) algorithm was obtained by La Poute [7] Our algorithm is based on a simple planarity test [5] developed by the author, which is a vertex addition algorithm based on a depth first search ordering. The ....

....to a postordering obtained from a depth first search tree. Given an undirected graph G, the maximal planar subgraph problem is to determine a planar subgraph H of G such that no edge of GH can be added to H without destroying planarity. Polynomial time algorithms have previously been given by [2,3,6,9] and a recent O(m a a a a a a a a a a a a a a a a a a a a a a a a a a a a (n) algorithm was obtained by La Porte [7] We solved the MPS problem in linear time by modifying the algorithm in [5] In the next section we briefly describe the idea in [5] Our MPS algorithm is discussed in Sections 3 ....

G. Di Battista and R. Tamassia [1989], Incremental planarity testing, in Proc. 30th Annual IEEE Symposium on Foundation of Computer Science, 436-411.

.... and Cederbaum planarity testing algorithm [10] Cai, Han, and Tarjan [1] proposed an O( E log V ) algorithm for finding a maximal planar subgraph based on the Hopcroft and Tarjan planarity testing algorithm [7] Another O( E log V ) algorithm was also proposed by Di Battista and Tamassia [2], based on an O(log V ) time per operation strategy to the problem of maintaining a planar graph under edge additions. Takefuji and Lee [15] described a parallel heuristic for planarization based on neural networks. They use an arbitrary sequencing of the vertices, placing them along a line, ....

G. Di Battista and R. Tamassia, Incremental planarity testing, in Proceedings of the 30th IEEE Symp. FOCS, Chapel Hill, NC, 1989, pp. 436--441.

.... Their result improved (if m = o(n 2 = log n) the best previous O(n 2 ) algorithm from [16] based on the PQ tree technique [2] An algorithm with the same complexity bound of O(m log n) can also be derived from the incremental planarity testing algorithm of Di Battista and Tamassia [5]. Using an approach similar to [5] Westbrook [23] described an algorithm that works in O(n log n mff(m; n) worstcase time plus an additional O(n) expected time. La Poutr e [21] recently gave an incremental planarity testing algorithm that takes O(ff(m; n) amortized time per operation, which ....

.... m = o(n 2 = log n) the best previous O(n 2 ) algorithm from [16] based on the PQ tree technique [2] An algorithm with the same complexity bound of O(m log n) can also be derived from the incremental planarity testing algorithm of Di Battista and Tamassia [5] Using an approach similar to [5], Westbrook [23] described an algorithm that works in O(n log n mff(m; n) worstcase time plus an additional O(n) expected time. La Poutr e [21] recently gave an incremental planarity testing algorithm that takes O(ff(m; n) amortized time per operation, which can be transformed into an O(n ....

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G. Di Battista, R. Tamassia, Incremental planarity testing, Proc. IEEE Symp. on Found. of Comp. Sci., (1989), 436-441.

.... the corresponding fully dynamic problems [11] Moreover, despite intensive research on dynamic problems on graphs (such as dynamic maintenance of connectivity [7, 8, 10, 11, 14, 20, 22, 29, 30] 2 and 3 connectivity [7, 12, 29, 30] transitive closure [3, 4, 15, 16, 17, 18, 19, 31] planar graphs [6, 7, 19, 25], shortest paths [2, 9, 21, 24, 31] and minimum spanning trees [5, 8, 11, 24] there are very few graphtheoretic problems for which a fully dynamic non trivial algorithm is known. As mentioned in [30] the fully dynamic maintenance of the connected components of a graph differs substantially from ....

G. Di Battista, and R. Tamassia, "Incremental planarity testing", Proc. 30th Annual Symp. on Foundations of Computer Science, 1989, 436--441.

.... forests [Wes89] shortest paths [Rod68, Che76, GSV78, Fuj81, CC82, Gaz83, EG85, AMSN89, AIMSN90, Ita91] biconnected components [Sac86, WT92, BT90] triconnected components [Ita91, BT90] transitive closure [IK83, Ita86, Ita88, LPv88, YS88, Yel91] planar graphs [Tam88, TP90, BT89, EIT 92, PT88] ffl computational geometry [Ov81, CBT 92] ffl data bases [ABJ89] ffl syntax directed editors and grammars [Rep82, RTD83, Rep88, ACR 87] ffl data flow analysis [Ryd83, RP88, RC86, CR88, Mar89, Bur90, PS89, Zad84, Gho83, KRvM88] ffl code generation and optimization ....

G. Di Battista and R. Tamassia. Incremental planarity testing. In Proceedings of the Thirtieth Annual IEEE Symposium on the Foundations of Computer Science, pages 436--441. Institute of Electrical and Electronics Engineers -- Computer Society, 1989.

....Least Common Ancestor return the least common ancestor of two tree nodes. ffl Find Minimum Find the minimum weight node on a path. Dynamic trees were introduced as internal data structures in sequential maximum flow algorithms [24,55,100] Since then, a large number of dynamic algorithms [27,28, 48,49,50,51,69,83,122] have used dynamic trees as part of their data structures. Initial data structures [24,55] based on balanced binary trees (e.g. AVL trees [1] or Red Black trees [57] take O(log 2 n) time per operation. Sleator and Tarjan improve this to O(log n) time per operation by basing their data ....

....structure with O( p m) time per update and O(m) space. When only considering planar graphs, the time bound is reduced to O(log 2 n) time per update. For fixed embeddings of planar graphs, this result is improved to O(log n) time per dynamic operation in [39] The semi dynamic data structure of [27] for embedding independent, biconnected planar graphs uses O(n) space and supports insertions in O(log n) time (amortized for edge insertions) 2.6 Planarity Testing In a static environment we can test planarity and compute a planar embedding in optimal O(n) time (see, e.g. 13,42,47,60,72] The ....

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G. Di Battista and R. Tamassia, "Incremental Planarity Testing," Proc. 30th IEEE Symp. on Foundations of Computer Science (1989), 436--441.

....problem, in which one must test if a given graph is planar, has been studied since the nineteen sixties. If can be solved in O(n) sequential time [3, 12] and O(log n) time on a CRCW PRAM [17] Recently Di Battista and Tamassia introduced and studied the incremental planarity testing problem [1, 2]: beginning with a biconnected graph G process on line an intermingled sequence of the following query and update operations. 1. Test(u; v) Determine if an edge can be inserted between vertices u and v of while still preserving the planarity of G. 2. Insert Vertex (v; e) Split edge e into two ....

....nodes that are children of the same R node. In this case, these two children are merged into one by equating the poles of the respective graphs. After phase 3, vertices u and v are incident on a common face of some triconnected graph and the edge is inserted. Figure 4 shows an example. Lemma 2. 1 [1] The total number of transformation steps over any sequence is O(n) There are O(n) marks made. To implement the test and update algorithms we need two fast data structures. The first supports the following operations on fixed embeddings. Test whether two vertices are adjacent in a fixed ....

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G. D. Battista and R. Tamassia. Incremental planarity testing. In Proc. 30th IEEE Symp. on Foundations of Computer Science, pages 436--441, 1989.

....Universitat zu Koln, Pohligstr. 1, 50969 Koln, Germany. E mail: leipert informatik.unikoeln. de P. Mutzel is with the Max Planck Institut fur Informatik, Im Stadtwald, 66123 Saarbrucken, Germany. E mail: mutzel mpi sb.mpg.de based on the path addition algorithm as well. Di Battista and Tamassia [4] described an algorithm that checks in O(log n) amortized time, whether an edge can be added to G without destroying planarity, obtaining an O(m log n) time algorithm as well. Ozawa and Takahashi [16] have presented an O(nm) algorithm using the vertex addition algorithm. Jayakumar, Thulasiraman ....

G. Di Battista and R. Tamassia. Incremental planarity testing. In Proceedings on the 30th Annual IEEE Symposium on Foundations of Computer Science, North Carolina, pages 436--441, 1989.

....a maximal planar subgraph containing it. Cai, Han, and Tarjan [3] proposed an O(jEj log jV j) algorithm for finding a maximal planar subgraph based on the Hopcroft and Tarjan planarity testing algorithm [14] Another O(jEj log jV j) algorithm was also proposed by Di Battista and Tamassia [7], based on an O(log jV j) time per operation strategy to the problem of maintaining a planar graph under edge additions. July 16, 1996 Latest version can be found in URL = ftp: netlib.att.com math people mgcr doc gmpsg.ps.Z y Information Sciences Research Center, AT T Research, Murray ....

G. Di Battista and R. Tamassia, Incremental planarity testing, in Proceedings of the 30th IEEE Symp. FOCS, Chapel Hill, NC, 1989, pp. 436--441.

....that s and t are on the boundary of the external face. Planar st graphs were first introduced by Lempel, Even, and Cederbaum [77] in connection with a planarity testing algorithm, and have subsequently been used in a host of applications, dealing with partial orders [68] planar graph embedding [30, 42, 111], graph planarization [91] graph drawing [41, 43] floor planning [123] planar point location [49, 94] visibility [73, 88, 102, 113, 114, 124] motion planning [101] and VLSI layout compaction [123] In this section, we consider a planar st digraph G with V vertices, and recall that G has ....

G. Di Battista and R. Tamassia. Incremental planarity testing. In Proc. 30th Annu. IEEE Sympos. Found. Comput. Sci., pages 436--441, 1989.

.... with preassigned order on one layer NP hard [43] general graph compute maximum planar subgraph NP hard [53] general graph planarity testing and computing a planar embedding O(n) Omega Gamma n) 8, 13, 47, 22, 68, 82] general graph compute maximal planar subgraph O(n m) Omega Gamma n m) [32, 62, 80, 36] general digraph upward planarity testing NP hard [60] embedded digraph upward planarity testing O(n 2 ) Omega Gamma n) 3] single source digraph upward planarity testing O(n) Omega Gamma n) 4, 69] general graph draw as the intersection graph of a set of unit diameter disks in the plane ....

....and deletions of vertices and edges. Trade offs between running time, optimization of the drawing properties, and preservation of the mental map are typical issues to be addressed in dynamic graph drawing. Most of the existing results on dynamic graph drawing are limited to planar graphs [15, 32, 45, 80, 89]. EXPERIMENTATION Many graph drawing algorithms have been implemented and used in practical applications. Most papers show sample outputs, and some also provide limited experimental results on small test suites. However, in order to evaluate the practical performance of a graph drawing algorithm ....

G. Di Battista and R. Tamassia. Incremental planarity testing. In Proc. 30th Annu. IEEE Sympos. Found. Comput. Sci., pages 436--441, 1989.

....techniques appears instead to be lacking in the area of dynamic graph algorithms. The goal of this research is exactly to provide such generalized techniques in the realm of dynamic tree and graph problems. Our approach is motivated by the observation that a number of dynamic graph algorithms [10,11,13,14,15,16,19,23,31], developed mostly for connectivity problems, appear to be based on the following fundamental idea: Decompose a graph into subgraphs with limited overlap, and represent such a decomposition by means of a tree so that dynamic operations on the graph are reflected into corresponding dynamic tree ....

G. Di Battista and R. Tamassia, "Incremental Planarity Testing," Proc. 30th IEEE Symp. on Foundations of Computer Science (1989), 436--441.

....is not monotone. Namely, there exists a sequence of insertions of vertices and edges on a planar graph G such that G alternatively gains or loses (strictly ) convex planarity after each insertion operation. Previous work on dynamic planarity testing and dynamic graph drawing is reported in [3, 4, 11, 12, 13, 14, 17, 18, 25, 40]. Besides their theoretical significance, our results are motivated by the development of advanced graph drawing systems. A variety of visualization applications require to automatically draw graphs. Examples include programming environments (e.g. displaying entity relationship diagrams and ....

....lowest common ancestor of two nodes) in logarithmic time. As shown in [18] they can be modified to support ordered trees and expand contract operations. In the description of time bounds we use standard concepts of amortized complexity [34] In the rest of this section, the SPQR tree presented in [11, 12, 13, 14] is described. Let G be a biconnected graph. A split pair of G is either a pair of adjacent vertices or a separation pair. In the former case the split pair is said trivial, in the latter non trivial. A split component of a split pair fu; vg is either an edge (u; v) or a maximal subgraph C of G ....

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G. Di Battista and R. Tamassia. Incremental planarity testing. In Proc. 30th Annu. IEEE Sympos. Found. Comput. Sci., pages 436--441, 1989.

....from s and t is mapped to its unique proper node. Each node is mapped to the source and sink of its pertinent digraph. Additionally, each S node is the proper node of a unique vertex. The concepts of proper node and skeleton are analogous to the corresponding ones developed for planar st graphs in [10]. We equip the tree T with a secondary data structure for maintaining the S chains and their skeletons. Namely, we store with each S chain Gamma = 1 ; 2 ; Delta Delta Delta ; r Gamma1 ) the poles v 0 and v r of its skeleton and a balanced binary tree, called routing tree. The i th (1 ....

G. Di Battista and R. Tamassia, "Incremental Planarity Testing," Proc. 30th IEEE Symp. on Foundations of Computer Science (1989), 436--441.

....28, 34] Dynamic planar graphs arise in communication networks, graphics, and VLSI design, and they occur in algorithms that build planar subdivisions such as Voronoi diagrams. Algorithms have been proposed for maintaining the embedding of a planar graph [29] and for incremental planarity testing [2, 3]. The dynamic minimum spanning tree problem has been considered by Spira and Pan [28] Chin and Houck [7] Frederickson [10] and Gabow and Stallmann [11] Frederickson gives an algorithm based on topology trees that runs in O( p m) time per operation on general graphs, and O( log n) 2 ) ....

....forest in O(1) time, and to determine the spanning tree containing a given vertex, or find the edge of maximum or minimum weight on the tree path between two vertices, in O(log m) time. The edge ordered tree also finds use in the on line planarity testing algorithm of Di Battista and Tamassia [2, 3]. Thus our data structure is fairly general and powerful. The algorithms can be made to run in worst case time O(log m) with the biased tree implementation of dynamic trees [26] We also argue that in our machine model, any algorithm must spend Omega Gammad 1 n) amortized time per operation; we ....

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G. D. Battista and R. Tamassia. Incremental planarity testing. In Proc. 30th IEEE Symp. on Foundations of Computer Science, pages 436--441, 1989.

....planar graphs. Planar st digraphs, which include series parallel digraphs as a special case, were first introduced by Lempel, Even, and Cederbaum [27] in connection with a planarity testing algorithm, and they have subsequently been used in several applications, including planar graph embedding [4,8,42], graph drawing [7,9] and planar point location [16, 21,32,48] A planar st digraph is a planar acyclic directed graph with exactly one source vertex s and exactly one sink vertex t, which is embedded in the plane such that s and t are on the boundary of the external face. The following ....

....v 0 ; v 00 ; edge e; face f; f 0 ; f 00 ) Add edge e = v 0 ; v 00 ) inside face f , which is decomposed into faces f 0 and f 00 . InsertVertex (vertex v; edge e; e 0 ; e 00 ) Insert vertex v on edge e, which is decomposed into edges e 0 and e 00 . As shown in [8], this repertory of operation is complete; i.e. any n vertex biconnected planar graph can be assembled by means of O(n) operations of the repertory. In the rest of this section we prove the following theorem: Theorem 5 Consider the following dynamic graph drawing problem: ffl Class of graphs G: ....

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G. Di Battista and R. Tamassia, "Incremental Planarity Testing," Proc. 30th IEEE Symp. on Foundations of Computer Science (1989), 436--441.

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