N. Chiba, T. Nishizeki, S. Abe, and T. Ozawa. A linear algorithm for embedding planar graphs using PQ-trees. Journal of Computer and System Sciences, 30:54-- 76, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... Proof of Lemma 2. We rst give a simple algorithm to construct a well orderly embedding of G. Then we give some hints for an O(n) time implementation. We start by computing an arbitrary plane embedding H of G such that v belongs to the outerface of H. This can be done in O(n) time [CNAO85] Then we traverse H from v in order to build a well orderly tree T . However, not any plane embedding allows the construction of a well orderly tree. If during the construction, T does not span all the nodes, the embedding of H is modi ed, and a new traversal is run again. We show that, after ....

Norishige Chiba, Takao Nishizeki, Shigenobu Abe, and Takao Ozawa. A linear algorithm for embedding planar graphs using pq-trees. Journal of Computer and System Sciences, 30(1):54-76, 1985.

....nition of i. Proof of Lemma 2. We rst give a simple algorithm to construct a well orderly embedding of G. Then we give some hints for an O(n) time implementation. We start by computing an arbitrary plane embedding H of G such that v belongs to the outerface of H. This can be done in O(n) time [CNAO85] Then we traverse H from v in order to build a well orderly tree T . However, not any plane embedding allows the construction of a well orderly tree. If during the construction, T does not span all the nodes, the embedding of H is modi ed, and a new traversal is run again. We show that, after ....

Norishige Chiba, Takao Nishizeki, Shigenobu Abe, and Takao Ozawa. A linear algorithm for embedding planar graphs using pq-trees. Journal of Computer and System Sciences, 30(1):54-76, 1985.

....2 for details. Degree: the degree of node v i is the sum of the degree in the tree T (see Lemma 4) the cardinality of B (v i ) see Lemma 5) and of B (v i ) has a front edge. Starting from a connected outerplanar graph, we can compute a rooted outerplanar map using the algorithm presented in [CNAO85]. So the previous results on outerplanar maps can also be applied to outerplanar graphs. 5 Uniform Random Generation To randomly generate an outerplanar map, one can randomly generate a bicolored tree where the nodes of the last branch are colored white. Thanks to Corollary 1, one can then ....

Norishige Chiba, Takao Nishizeki, Shigenobu Abe, and Takao Ozawa. A linear algorithm for embedding planar graphs using pq-trees. Journal of Computer and System Sciences, 30(1):54-76, 1985.

....undirected graph G = V; E) consists of constructing adjacency lists A(v) for each node v 2 V , such that all the neighbors of v appear in clockwise order with respect to a planar drawing of G. Such a set of adjacency lists is called a (combinatorial) embedding of G. In 1985, Chiba and Nishizeki [CN] presented an algorithm for constructing a combinatorial embedding based on the PQ tree approach. They wrote: Hopcroft and Tarjan mentioned that an embedding algorithm can be constructed by modifying their testing algorithm. However, the modification looks to be fairly complicated; in particular, ....

Chiba, N. and T. Nishizeki, A Linear Algorithm for Embedding Planar Graphs Using PQ-Trees, J. of Comp. and Sys. Sci., vol. 30, pp. 54-76,1985

....work related to our combinatorial results. Bipolar orientations and st numberings of biconnected graphs were first defined in conjunc tion with a planarity testing algorithm [18, 33] and were later used for a variety of topological and geometric graph problems, such as embedding (see, e.g. [6, 15, 42]) visibility (see, e.g. 36, 43, 53] drawing (see, e.g. 1, 12, 44] point location (see, e.g. 35, 45] and floorplanning (see, e.g. 30] One of the notable properties of planar bipolar orientations is that they induce a 2 dimensional lattice [31] on the vertices of the graph. See [10] ....

N. Chiba, T. Nishizeki, S. Abe, and T. Ozawa. A linear algorithm for embedding planar graphs using PQ-trees. J. Cornput. Syst. Sci., 30(1):54 76, 1985.

....destroying planarity. A maximum 2 planar subgraph has maximum cardinality among all planar subgraphs of G. Several efficient methods have been developed for testing graph planarity. Perhaps the two best known methods are those of Hopcroft and Tarjan [16] and Booth and Lueker [4] also see [1]) Both have linear time complexity in the number of vertices of the graph. With either method, however, no attempt is made to proceed after nonplanarity is detected. Hence, the methods must be extended to find a maximum planar subgraph. In the next section we discuss heuristics for finding large ....

S. Abe, N. Chiba, T. Nishizeki, and T. Ozawa, A linear algorithm for embedding planar graphs using PQ-trees, J. Comput. Sys. Sci., 30 (1985), 54-76.

....The distinct phases of the topology shape metrics approach have been extensively studied in the literature. If G is planar (which can be tested in linear time with one of the well known algorithms in [25, 29] an embedding of G is determined in linear time, by applying an embedding algorithm [14, 30]. If G is not planar, the minimum number of dummy vertices introduced may be n 4 ) However, in practice this number is usually much smaller. Minimizing the number of crossings is in general NP hard [21] For a survey on planarization techniques see [17] A popular algorithm for constructing ....

N. Chiba, T. Nishizeki, S. Abe, and T. Ozawa. A linear algorithm for embedding planar graphs using PQ-trees. J. Comput. Syst. Sci., 30(1):54-76, 1985.

....separators is well studied. We mention the most relevant results here. In their classic paper [21] Lipton and Tarjan show that a 2 3 separator of size O( p N) can be computed in linear time for a given embedded planar graph G of size N. The required embedding can be computed in linear time [17, 7, 15, 20, 11, 24]. In [16] the algorithm of [21] is applied recursively to compute in O(N logN) time a set S of O(N=h) vertices whose removal partitions G into O(N=h) possibly disconnected) subgraphs of size O(h) each of which is adjacent to at most O( p h) vertices in S. In [3] it is shown how to compute a ....

N. Chiba, T. Nishizeki, S. Abe, and T. Ozawa. A linear algorithm for embedding planar graphs using PQ-trees. Journal of Computer and Systems Science, 30(1):54--76, 1985.

....m=2 and therefore n 4 p 2m. Then by Corollary 1.1, we have (G) Kn ) b(n 9) 6c b p 2m=3 3=2c. 4. A THICKNESS HEURISTIC There are O(n) algorithms for testing graph planarity. Perhaps the two bestknown methods are those of Hopcroft and Tarjan [13] and Booth and Lueker [5] also see [1]) Either method must be extended to compute the thickness. One approach is to repeatedly extract maximal planar subgraphs from a nonplanar graph until the resulting subgraph is planar. If the subgraphs extracted are nearly optimal in size then a fairly good approximation to thickness should ....

S. Abe, N. Chiba, T. Nishizeki, and T. Ozawa. A linear algorithm for embedding planar graphs using PQ-trees. J. Comput. Sys. Sci. 30:54-76 (1985).

....of an e#cient algorithm for determining the genus of graphs of bounded tree width. It turned out that some of the main ingredients in this proof can also be found in the aforementioned work of Seymour [55] It is well known that testing planarity [20] constructing embeddings in the sphere S 0 [12], or finding subgraphs that are subdivisions of Kuratowski graphs [62] can be performed by algorithms whose worst case running time is linear. Although the construction of minimum genus embeddings is NP hard (by Minors and embeddings 11 Thomassen [58] Filotti, Miller, and Reif [16] proved that ....

N. Chiba, T. Nishizeki, S. Abe, T. Ozawa, A linear algorithm for embedding planar graphs using PQ-trees, J. Comput. System Sci. 30 (1985) 54--76.

....(see Figure 6) to get a VAP free planar representation of G 1 . 2 The characterization obtained, allows us to get an algorithm testing efficiently whether a dynamic graph is VAP free planar or not because algorithms to test if a graph is planar appear in [14] and [3] Moreover, N. Chiba et al. [4] give 9 an algorithm that embeds a finite planar graph in the plane in linear time (see [20] so we can get an efficient embedding of G 1 without vertex accumulation points. As a consequence, we have that Theorem 4 Let G 1 be a dynamic graph. Then, it can be computed whether or not G 1 is ....

N. Chiba, T. Nishizeki, S. Abe and T. Ozawa, A linear algorithm for embedding planar graphs using PQ-trees, J. Comput. Syst. Sci. 30(1) (1985) 54--76.

....can have different planar embeddings. This statement also holds for rectilinear embeddings, since any planar embedding with curved edges can be transformed into a rectilinear embedding [1] In the literature a planar embedding is specified with the ordered adjacency lists of its vertices (see [5]) It is immediate to verify that this description of a line drawing is stable within the viewing regions and changes only crossing their boundaries. It is also easily seen that the ordered adjacency lists are different for the pairs of line drawings in Fig. 2 and Fig. 3. Unfortunately however, as ....

N. Chiba et al., A linear algorithm for embedding planar graphs using PQ-trees, J. Comput. System Sci. 30 (1985) 54--76.

....Using this result, and introducing a data structure called PQ trees, Booth and Lueker [BL76] improved Lempel, Even, and Cederbaum s planarity testing algorithm to run in linear time. The algorithm was modified to also yield a combinatorial embedding for the graph if it is planar by Chiba et al. CNAO85] Eve79, Section 8.4] and [TS92, Section 11.11] describe the original algorithm [LEC67] and [Kan93, Section 2.2.2] describes the implementation [BL76] using PQ trees. Recently, two di#erent, new, planarity testing and embedding algorithms have been proposed [SH99, BM99] 1.3 Generalizations ....

Norishige Chiba, Takao Nishizeki, Shigenobu Abe, and Takao Ozawa. A Linear Algorithm for Embedding Planar Graphs Using PQ-Trees. J. Computer and System Sciences, 30:54--76, 1985.

.... general graph minimize crossings NP hard [54] 2 layered graph minimize crossings in layered drawing 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, ....

N. Chiba, T. Nishizeki, S. Abe, and T. Ozawa. A linear algorithm for embedding planar graphs using PQ-trees. J. Comput. Syst. Sci., 30(1):54--76, 1985.

.... by Hopcroft and Tarjan [9] There are several other linear time planarity algorithms (e.g. Booth and Lueker [1] Fraysseix and Rosenstiehl [6] Williamson [23, 24] Extensions of original algorithms produce also an embedding (rotation system) whenever the given graph is found to be planar [2], or find a small obstruction a subgraph homeomorphic to K 5 or K 3;3 if the graph is non planar [23, 24] The subgraph homeomorphic to K 5 or K 3;3 is called a Kuratowski subgraph of G. Kuratowski subgraph Lemma 2.1 There is a linear time algorithm that, given a graph G, either exhibits ....

N. Chiba, T. Nishizeki, S. Abe, T. Ozawa, A linear algorithm for embedding planar graphs using PQ-trees, J. Comput. System Sci. 30 (1985) 54--76.

....differently with planar and non planar graphs. The case of planar graphs is simpler, and in it we carry out the following: Phase A: Planarity testing. Phase B: Planar embedding. Phase C: Planar drawing. Phase D: Randomized beautification. Phase B uses the PQ trees based algorithm presented in [CNAO85] to construct a planar embedding, i.e. an ordered list of the neighbors of each vertex, which, if layed out appropriately in cyclic order around the vertex, leads to a planar drawing. Phase C then uses the embedding lists produced by the previous phase to actually draw the graph. The output is a ....

....a brief description of the drawing algorithm for planar graphs, that constitutes phase C of our system. A more detailed description can be found in [Sar93, HS93] The input to this phase is a planar graph accompanied by the planar embedding constructed in phase B using the PQ trees algorithm of [CNAO85]. The output is a planar drawing of the graph that complies with the given embedding. By a planar embedding of a graph G we mean an array of lists, one for each vertex, with v s list containing the edges incident to it in circular order around v in a possible planar drawing of G. Our algorithm is ....

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Chiba, N., T. Nishizeki, S. Abe, and T. Ozawa, "A Linear Algorithm for Embedding Planar Graphs Using PQ-trees", J. Comput. Syst. Sci. 30:1 (1985), 54--76.

.... some new applications [17] Another planarity test algorithm based on work of Lempel, Even, and Cederbaum [12] was optimized to run in linear time by using Even and Tarjan s s; t numbering scheme [6] together with the PQ tree data structure of Booth and Lueker[2] Chiba, Nishizeki, Abe and Ozawa [3] use this approach to create a linear time planar embedding algorithm. The 1985 algorithm of de Fraysseix and Rosentiehl [4] should also be noted in regards to this 1 2 Figure 1: The planar obstructions K 5 and K 3;3 (a) K (b) K 5 3,3 research because their approach is also based on the ....

N. Chiba, T. Nishizeki, A. Abe, and T. Ozawa. A linear algorithm for embedding planar graphs using PQ--trees. J. Comput. Sys. Sci., 30:54--76, 1985.

....gave the first complete characterization of planar graphs. Unfortunately Kuratowski s characterization does not lead to an e#cient algorithm for planarity testing. Linear time algorithms for this problem have been developed by Hopcroft and Tarjan [HT74] and Booth and Lueker [BL76] Chiba et al. [CNAO85], Cai et al. CHT89] and Mutzel [Mut92] gave linear time algorithms for finding a planar representation of a planar graph. A planar graph with a fixed planar representation is called a plane graph. 1.2.3 Straight line drawings A straight line drawing is a drawing of a graph in which each edge of ....

N. Chiba, T. Nishizeki, S. Abe and T. Ozawa, A linear algorithm for embedding planar graphs using PQ-trees, J. Comput. Syst. Sci., 30 (1985) pp. 54-76.

....in 1976 partly by Even and Tarjan [4] and partly by Booth and Lueker [1] The P Q tree approach is conceptually simpler, but its implementation is more complicated than that of the H T algorithm. Linear time solutions for P3 and P4, also based on P Q trees, were given by Chiba et al. [2] in 1985. Wu [15] gave an algebraic solution for all four problems. He proved that a graph is planar if and only if a certain system of linear equations is solvable. In case the graph is planar, an actual embedding can be obtained by considering another system of quadratic equations. His solution ....

....not empty, then the vertex low 1 (e) is always on cycle (e) Also, if low 1 (e) a, then sub (e) S (e) if low 1 (e) a, then sub (e) S (e) e: e belongs to the tree path from low 1 (e) to a . Fig. 6 illustrates some of these definitions, where low 1 (e) 1; low 2 (e) 2; cycle (e) [1, 2], 2, 3] 3, 4] 4, 5] 5, 6] 6, 7] 7, 8] 8, 1] S (e) contains all the edges in the graph except [1, 2] 2, 3] 3, 4] sub (e) is the whole graph; ATT (e) 8, 1] 9, 3] 12, 1] 14, 2] 13, 4] 6 Fig. 6 tree edge back edge 7 6 9 8 e 5 4 3 2 1 10 11 ....

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Chiba, N., Nishizeki, T., Abe, S., and Ozawa, T., "A Linear Algorithm for Embedding Planar Graphs Using PQ-trees," Journal of Computer and System Sciences, vol. 30, no. 1, pp. 54-76, 1985.

.... Counting Embeddings of Planar Graphs Using DFS Trees 1 Jiazhen Cai Courant Institute, NYU New York, NY 10012 ABSTRACT Previously counting embeddings of planar graphs [5] used P Q trees and was restricted to biconnected graphs. Although the P Q tree approach is conceptually simple, its implementation is complicated. In this paper we solve this problem using DFS trees, which are easy to implement. We also give formulas that count the number of embeddings of ....

.... 2 Wu solved all these problems using systems of algebraic equations. His solutions are elegant, but his implementations are not so efficient. Other solutions to these problems basically follow two different approaches. One uses DFS trees [4, 8] and the other uses P Q trees [3, 5, 9 11]. The P Q tree approach is considered to be conceptually simpler, but its implementation is much more complicated. Efficient P Q tree solutions have been discovered for all the four problems. Lempel, Even and Cederbaum [10] solved problem 1. Chiba et al. solved problems 3 and 4 [5] These ....

[Article contains additional citation context not shown here]

Chiba, N., Nishizeki, T., Abe, S., and Ozawa, T., "A Linear Algorithm for Embedding Planar Graphs Using PQ- trees," Journal of Computer and System Sciences, vol. 30, no. 1, pp. 54-76, 1985.

....time. This data structure has been introduced by Booth and Lueker (1976) to solve the problem of testing for consecutive ones property. The most well known applications of PQ trees in Automatic Graph Drawing are planarity testing (see Lempel et al. 1967; Booth and Lueker, 1976) and embedding (see Chiba et al. 1985). Both are difficult to implement but very efficient, therefore PQ trees have become standard tools in automatic graph drawing systems. Other attempts to use algorithms based on PQ trees for automatic graph drawing problems have not been successful. One well known example is the computation of ....

Chiba, N., Nishizeki, T., Abe, S., and Ozawa, T. (1985). A linear algorithm for embedding planar graphs using PQ-trees. Journal of Computer and System Sciences, 30, 54--76.

....is obviously bounded by O(jEj) Hence the total number of operations is bounded by O(jV j) 2 4 Remarks An embedding of a level planar graph G = V; E) can also be computed in linear time. We apply our level planarity test with additional PQ tree techniques as they have been described by Chiba et al. 1985]. These techniques are based on two main ideas. One is to compute for every vertex v 2 V j , j 2 f2; 3; kg a permutation of its neighbors on level j Gamma 1 that appears in a level planar embedding. Knowing a permutation of the k level vertices in a level planar embedding of G, we can ....

N. Chiba, T. Nishizeki, S. Abe, and T. Ozawa. A linear algorithm for embedding planar graphs using PQ-trees. Journal of Computer and System Sciences, 30:54--76, 1985.

....test and augments a level planar graph G to an st graph G st , a graph with a single sink and a single source, without destroying the level planarity. Once the st graph has been constructed, we compute a planar embedding of the st graph. This is done by applying the embedding algorithm of Chiba et al. 1985] for general graphs, obeying the topological ordering of the vertices in the st graph. Exploiting the embedding of the st graph G st , we are able to determine a level planar embedding of G. This extended abstract is organized as follows. After summarizing the necessary preliminaries in the next ....

....every v; w 2 V st we have ts G st (v) ts G st (w) if and only if lev G st (v) lev G st (w) Obviously ts G st is an st numbering of G st . Using this st numbering, we can obtain a planar embedding E st of G st with the edge (s; t) on the boundary of the outer face by applying the algorithm of Chiba et al. 1985]. From the planar embedding we obtain a level planar embedding of G st by applying a function CONSTRUCT LEVEL EMBED that uses a depth first search procedure starting at vertex t and proceeding from every visited vertex w to the unvisited neighbor that appears first in the clockwise ordering of ....

[Article contains additional citation context not shown here]

N. Chiba, T. Nishizeki, S. Abe, and T. Ozawa. A linear algorithm for embedding planar graphs 9 using PQ-trees. Journal of Computer and System Sciences, 30:54--76, 1985.

....adding an outgoing edge to every sink without destroying level planarity. 2. Add an extra vertex s on an extra level 0 and compute an st graph by adding the edge (s; t) and an incoming edge to every source without destroying the level planarity. 3. Compute a planar embedding using the algorithm by Chiba et al. 1985]. 4. Construct a level planar embedding from the planar embedding. The difficult part is to insert edges without destroying level planarity. We apply the following strategy (see also Leipert [1998] The idea is to determine the position of a sink t 2 V j , j 2 f1; 2; k Gamma 1g by ....

N. Chiba, T. Nishizeki, S. Abe, and T. Ozawa. A linear algorithm for embedding planar graphs using PQ-trees. Journal of Computer and System Sciences, 30: 54--76, 1985.

....work related to our combinatorial results. Bipolar orientations and st numberings of biconnected graphs were first defined in conjunction with a planarity testing algorithm [18, 33] and were later used for a variety of topological and geometric graph problems, such as embedding (see, e.g. [6, 15, 42]) visibility (see, e.g. 36, 43, 53] drawing (see, e.g. 1, 12, 44] point location (see, e.g. 35, 45] and floorplanning (see, e.g. 30] One of the notable properties of planar bipolar orientations is that they induce a 2 dimensional lattice [31] on the vertices of the graph. See [10] ....

N. Chiba, T. Nishizeki, S. Abe, and T. Ozawa. A linear algorithm for embedding planar graphs using PQ-trees. J. Comput. Syst. Sci., 30(1):54--76, 1985.

....and Tarjan [HT74] gave an algorithm that tests the planarity of an undirected graph in linear time. Alternative linear time algorithms were developed by Lempel, Even, and Cederbaum [LEC67, ET76] Booth and Lueker [BL76] and Fraysseix and Rosenstiehl [FR82] Chiba, Nishizeki, Abe and Ozawa [CNAO85] have shown how to extend the algorithm of Booth and Lueker so as to also construct a planar combinatorial embedding. Hopcroft and Tarjan also stated but gave no details that their planarity testing algorithm can be extended to also construct a planar combinatorial embedding. The textbook of the ....

N. Chiba, T. Nishizeki, S. Abe, and T. Ozawa. A linear algorithm for embedding planar graphs using PQ-trees. J. of Computer and System Sciences, 30(1):54--76, 1985.

....This data structure has been introduced by Booth and Lueker (1976) to solve the problem of testing for the consecutive ones property. The most well known applications of PQ trees in automatic graph drawing are planarity testing (see Lempel et al. 1967; Booth and Lueker, 1976) and embedding (see Chiba et al. 1985). Therefore PQ trees have become standard tools in automatic graph drawing systems. Other attempts to use algorithms based on PQ trees for automatic graph drawing problems have not been successful. One well known example is the computation of maximal planar subgraphs. Given a simple, connected ....

Chiba, N., Nishizeki, T., Abe, S., and Ozawa, T. (1985). A linear algorithm for embedding planar graphs using PQ-trees. Journal of Computer and System Sciences, 30, 54--76.

....This method underlies both main types or approaches for planarity testing algorithms, path addition (also called edge addition) and vertex addition. Although no embedding is explicitly exhibited by either approach, all information needed to do so is present during the course of the algorithm ([HT74, BL76, CNAO85, Joh98]) The rst linear planarity testing algorithm was based on path addition and is due to Hopcroft and Tarjan [HT74] The rst planarity testing algorithm based on the vertex addition approach that reached the linear time bound, is due to Booth and Lueker [BL76] and is built on the algorithm given ....

....and obstructions to planarity In chapter 3, a planarity testing algorithm using the PQ tree data structure was presented. Both this algorithm, the vertex addition algorithm of [LEC67] and [BL76] and the path addition algorithm of [HT74] have been used for other purposes than planarity testing ([CHT93, Cim95, JTS89, Kan92, CNAO85, Kar90]) This chapter will concentrate on one of these articles, Kar90] where a linear algorithm for explicitly identifying obstructions to planarity is given. This article formed the basis for our search for a new approach to a skewness heuristic. Given a non planar graph G, then a subgraph G # of ....

[Article contains additional citation context not shown here]

N. Chiba, T. Nishizeki, S. Abe, and T. Ozawa. A linear algorithm for embedding planar graphs using PQ-trees. J. Comput. System Sci., 30(1):5476, 1985.

....of a planar embedding. A planar embedding is not an actual drawing of the graph, but is a data structure that describes the circular ordering of neighbors of each vertex in some planar drawing. A linear time algorithm for constructing a planar embedding for a planar graph is described in [CNAO85] (see also [Mel96] and it can be used as a preliminary step to the canonical ordering algorithm now described. The ordering algorithm works by processing each of the vertices (in an order explained below) and visiting its neighbors. Vertices are labeled, and when visiting neighbors the labels ....

....v more than once. Note that to construct the boundary lists we do not need the planar drawing itself; all we need is a planar embedding, as we only use the circular ordering of neighbors around each vertex. As mentioned earlier, a planar embedding can be found by the linear time algorithm of [CNAO85]. 3.2 The biconnected canonical ordering Let G be a biconnected planar graph drawn in the plane. Let G k be a connected subgraph of G, and let C k = w 1 ; w 2 ; wm be the counter clockwise boundary list of the exterior face of G k (we call C k the contour of G k ) Let v be a vertex in G ....

Chiba, N., T. Nishizeki, S. Abe, and T. Ozawa, "A Linear Algorithm for Embedding Planar Graphs Using PQ-trees", J. Comput. Sys. Sci. 30 (1985), 54--76.

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N. Chiba, T. Nishizeki, S. Abe, and T. Ozawa. A linear algorithm for embedding planar graphs using PQ-trees. Journal of Computer and System Sciences, 30:54-- 76, 1985.

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N. Chiba, T. Nishizeki, S. Abe, and T. Ozawa. A linear algorithm for embedding planar graphs using PQ-trees. Journal of Computer and System Sciences, 30:54--76, 1985.

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N. Chiba, T. Nishizeki, S. Abe, and T. Ozawa. A linear algorithm for embedding planar graphs using PQ-trees. Journal of Computer and System Sciences, 30:54-- 76, 1985.

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Chiba, N., Nishizeki, T., Abe, S., Ozawa, T.: A Linear Algorithm for Embedding Planar Graphs Using PQ-Trees. Journal of Computer and System Sciences, 30 (1985) 54--76

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Norishige Chiba and Takao Nishizeki, "A linear algorithm for embedding planar graphs using PQ-trees", Journal of Computer and System Sciences 30 (1985), 54-76.

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N. Chiba, T. Nishizeki, S. Abe, and T. Ozawa. A linear algorithm for embedding planar graphs using PQ-trees. Journal of Computer and System Sciences, 30:54-- 76, 1985.

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N. Chiba, T. Nishizeki, S. Abe, and T. Ozawa. A linear algorithm for embedding planar graphs using PQ-trees. Journal of Computer and System Sciences, 30:54-- 76, 1985.

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N. Chiba, T. Nishizeki, S. Abe, and T. Ozawa. A linear algorithm for embedding planar graphs using PQ-trees. Journal of Computer and System Sciences, 30:54{ 76, 1985.

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Norishige Chiba, Takao Nishizeki, Shigenobu Abe, and Takao Ozawa. A linear algorithm for embedding planar graphs using pq-trees. Journal of Computer and System Sciences, 30(1):54-76, 1985.

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N. Chiba, T. Nishizeki, S. Abe, and T. Ozawa. A linear algorithm for embedding planar graphs using PQ-trees. Journal of Computer and System Sciences, 30:54-76, 1985.

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N. Chiba, T. Nishizeki, S. Abe, and T. Ozawa. A linear algorithm for embedding planar graphs using PQ-trees. Journal of Computer and System Sciences, 30:54--76, 1985.

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Norishige Chiba, Takao Nishizeki, Shigenobu Abe, and Takao Ozawa. A linear algorithm for embedding planar graphs using pq-trees. Journal of Computer and System Sciences, 30(1):54-76, 1985.

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N. Chiba, T. Nishizeki, S. Abe, and T. Ozawa. A linear algorithm for embedding planar graphs using PQ-trees. Journal of Computer and System Sciences, 30:54-- 76, 1985.

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N. Chiba, T. Nishizeki, S. Abe, and T. Ozawa. A linear algorithm for embedding planar graphs using PQ-trees. Journal of Computer and System Sciences, 30:54-76, 1985.

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N. Chiba, T. Nishizeki, A. Abe, and T. Ozawa. A linear algorithm for embedding planar graphs using PQ{trees. Journal of Computer and Systems Sciences, 30:54-76, 1985.

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N. Chiba, T. Nishizeki, S. Abe, and T. Ozawa. A linear algorithm for embedding planar graphs using PQ-trees. Journal of Computer and System Sciences, 30:54-- 76, 1985. 34

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N. Chiba, T. Nishizeki, A. Abe, and T. Ozawa. A linear algorithm for embedding planar graphs using PQ{trees. Journal of Computer and Systems Sciences, 30:54-76, 1985.

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N. Chiba, T. Nishizeki, S. Abe, and T. Ozawa. A linear algorithm for embedding planar graphs using pq-trees. J. of Comp. and Sys. Sci., 30(1):54-76, 1985.

No context found.

N. Chiba, T. Nishizeki, S. Abe, T. Ozawa. A linear algorithm for embedding planar graphs using PQ-trees. J. Comput. Syst. Sci., 30: 54--76, 1985

No context found.

Norishige Chiba, Takao Nishizeki, Shigenobu Abe, and Takao Ozawa. A linear algorithm for embedding planar graphs using pq-trees. Journal of Computer and System Sciences, 30(1):54-76, 1985.

No context found.

N. Chiba, T. Nishizeki, S. Abe, and T. Ozawa. A linear algorithm for embedding planar graphs using PQ-trees. J. of Uomputer and System Sciences, 30(1):54-76, 1985.

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