J. Hopcroft and R.E. Tarjan, Efficient planarity testing, J. ACM 21 (1974), 549--568.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of graphs for which graph isomorphism is polynomial, and where the test of membership in F is also polynomial. We restrict the operations so that if the input pomset is not in F , then the result of the operation is the empty pomset. The class of planar graphs satisfy these two requirements [HT74, HW74] We assume in the following that the operations are restricted to pomsets whose (transitively reduced) graph representation is planar. This guarantees tractability of the operations. Such a restriction might seem artificial, but it is not unusual in computer science. In robotics for ....

J.E. Hopcroft and R.E. Tarjan. Efficient planarity testing. J. of the ACM, 21:549-- 568, 1974.

....fundamental non planar graphs, and that planar graphs could contain neither K 3;3 nor K 5 as subgraphs, nor any subgraph homeomorphic to them. The Kuratowski graphs are given as Figure 2. Demoucron, Malgrange and Pertuiset formulated a planarity test based on subgraphs [DMP64] Hopcroft and Tarjan [HT74] developed a much more effective test: perform a depth first search on the graph, find a simple cycle, and rebuild the the graph by adding the rest of the paths to the cycle. The original cycle is planar, and paths can only be added to it if they maintain planarity. The HopcroftTarjan algorithm ....

.... algorithm, since in general, determining the genus of a given graph is NP complete [Tho89] That said, under certain conditions, determining the graph genus does not pose a problem: in the trivial case, this method can be used to find a linear space representation of a planar graph in linear time [HT74]. The requirement that an embedding be found means that the time complexity of the algorithm is impossible to quantify, and possibly too high to be practical if the genus is unbounded. Another disadvantage is that while this algorithm behaves quite predictably, its space complexity does depend on ....

[Article contains additional citation context not shown here]

J. Hopcroft and R. E. Tarjan. Efficient planarity testing. J. ACM, 21:549--568, 1974. 14, 34, 46

.... of the region) was studied in [1, 2, 8] A simple linear time algorithm for this version of the TPR problem was developed in these papers For the case in which none of the modules are on the boundary, Pinter [8] has suggested using the linear time planarity testing algorithm of Hopcroft and Tarjan [12]. His algorithm is quite complex. Marek Sadowska and Tarng [1] have considered the TPR problem and several variants which include fiippable modules and multi terminal nets. They develop a linear time algorithm for TPR which is based on module merging. In this paper, we develop, in section 3, ....

....Nets We shakl refer to the extension of TestingPlanar Routability or TPR to the case where some or al nets may have more than two pins as MTPR. The MTPR problem may be solved in linear time by mapping MTPR instances into graph planarity instances [3, 1] However, the known linear time agorithms [12] for graph planarity are complex and one is motivated to explore the possibility that simpler agorithms exist for MTPR (just as they do for TPR) Unfortunately, this is not the case. We show, in Theorem 2, that any agorithm for MTPR can be used to test graph planarity with no increase in ....

John Hopcroft and Robert Tarjan, "Efficient Planarity Testing," J. ACM, vol. 21, no. 4, pp. 549-568, 1974.

....e 2 E, the problem consists of finding the minimum number of edges whose deletion from a nonplanar graph gives a planar subgraph. In either case the problem is NP hard [GJ79] The problem can be solved in polynomial time if G is already planar, since planarity testing can be done in linear time [HT74]. If G = K n , the complete graph on n nodes, or G = K m;n , the complete bipartite graph on n m nodes, it is easy to construct a solution which contains 3n Gamma 6, resp. 2n Gamma 4 edges, and so the unweighted problem is solved in linear time. A related problem to the unweighted maximum ....

....the original cycle. Then the embeddings of the pieces are combined, if possible, to give an embedding of the entire graph. One may think of successively adding paths consisting of tree edges and one back edge at the end to a previously obtained partial embedding. For more details, see [M92] or [HT74]. In the following we describe some details of the branch and cut algorithm. Cutting plane generation The trivial inequalities are handled implicitly by the LP solver via lower and upper bounds. At the beginning we also add the inequality x(E) 3jV j Gamma 6, if it is violated, resp. x(E) 2jV ....

Hopcroft, J., and R.E. Tarjan, "Efficient planarity testing", J. ACM 21 (1974) 549--568

....complete for FP . Proof. This is the same as the proof of Corollary 5, except that the span P computation rejects any subgraph H which is not a tree. A more direct proof could be obtained by using a polynomial time canonical labelling algorithm for trees such as the one by Hopcroft and Tarjan [6]. ....

J. E. Hopcroft and R. E. Tarjan, Efficient planarity testing, Journal of the ACM 21 (1974) 549--568.

....directed edge at most once. 6.4.2 Corollary Every 3 connected planar graph G has a 3 spanning tree. Such a tree can be constructed in linear time and space. Proof: Embed G with one of the planarity algorithms that runs in linear time and space (for example the algorithm of Hopcroft and Tarjan [HT74] or Shih and Hsu [SH96] Fix one face to be the outer face W. Apply the algorithm presented before. 6.5 Extensions In this section we will present two results: i) we can choose in a circuit graph G C a vertex v that has to be a leaf in the 3 spanning tree of G C and (ii) every 3 connected ....

J. Hopcroft and R. E. Tarjan. Efficient planarity testing. Journal of the Association for Computing Machinery, 21:549--568, 1974.

....edges around each vertex. A planar drawing subdivides the plane into regions called faces; these faces are determined by the planar embedding alone. Whenever we speak of a planar graph, we assume that some (arbitrary) planar embedding has been fixed beforehand; this can be computed in linear time [6, 23, 29]. The dual graph G of a planar graph G is obtained by creating a vertex in G for every face in G, and adding an edge (F 1 ; F 2 ) in G for every edge e in G that is incident to the two faces F 1 and F 2 ; F 1 ; F 2 ) is called the dual edge of e. The planar embedding of the dual graph is ....

John Hopcroft and Robert Tarjan. Efficient planarity testing. Journal of the ACM, 21(4):549--568, October 1974.

....edges around each vertex. A planar drawing subdivides the plane into pieces called faces; these faces are determined by the planar embedding alone. Whenever we speak of a planar graph, we assume that some (arbitrary) planar embedding has been fixed beforehand; this can be computed in linear time [19, 24]. The dual graph G of a planar graph G is obtained by creating a vertex in G for every face in G, and adding an edge (F 1 ; F 2 ) in G for every edge e in G that is incident to the two faces F 1 and F 2 ; F 1 ; F 2 ) is called the dual edge of e. 3 Non planar Case 3.1 Frink s Proof of ....

J. E. Hopcroft and R. E. Tarjan. Efficient planarity testing. Journal of the ACM, 21(4):549--568, 1974.

....the algorithm of [28] can be used to test whether a given graph is planar. In internal memory, planarity testing and the problem of computing a planar embedding of a given graph G are well studied. The first paper to present a linear time algorithm for planarity testing and planar embedding is [16]. A previous algorithm of [21] was later made to run in linear time using results of [6, 12] Important implementation details of the algorithm of [16] are provided in [24] Any graph traversal can be used to identify the connected components of a graph in linear time. In [29] a linear time ....

....embedding of a given graph G are well studied. The first paper to present a linear time algorithm for planarity testing and planar embedding is [16] A previous algorithm of [21] was later made to run in linear time using results of [6, 12] Important implementation details of the algorithm of [16] are provided in [24] Any graph traversal can be used to identify the connected components of a graph in linear time. In [29] a linear time algorithm for finding the biconnected components of a graph is presented. In [15] the idea of [29] was extended to identify the triconnected components of ....

[Article contains additional citation context not shown here]

J. Hopcroft and R. E. Tarjan. Efficient planarity testing. Journal of the ACM, 21(4):549--568, 1974.

....shown to be NP complete. Some characteristics of special cases and related issues are discussed in [1,9,10] Based on these, 11] identifies one large class of directed graphs planar directed graphs for which testing membership in the class is efficiently decidable, e.g. using the algorithm of [6], and for which a polynomial time algorithm deciding if such a graph contains an even length cycle is presented. 3 Bounds on the Number of Possible Stable Extensions Given an argument system H(hX ; Ai) and a subset T of X, we denote by St(H; T) the total number of different stable extensions ....

J.E. Hopcroft and R.E. Tarjan. Efficient planarity testing. Journal of the A.C.M, 21:549--568, 1974.

....the plane with no two edges crossing. Planar graphs are important in the construction of circuits, where a crossed wire would cause a short circuit. Determining whether a graph is planar or not (from the internal representation of it) can be done in O(n) time, due to a very sophisticated algorithm [10]. It is also known that all planar graphs can be colored (colors assigned to the vertices of the graph so that no two adjacent vertices have the same color) with at most four colors and an efficient algorithm exists to 4 color a planar graph. But these are quite complex. The original (computer ....

J. Hopcroft and R. Tarjan (1974), Efficient Planarity Testing, J. ACM, vol. 21, no. 4, pp. 549-568

....no edges of G Gamma H can be added without 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 ....

J. Hopcroft and R.E. Tarjan, Efficient planarity testing, J. Assoc. Comput. Mach. 21 (1974), 549-568.

....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 ....

J. Hopcroft and R. E. Tarjan. Efficient planarity testing. Journal of the ACM, 21(4):549--568, 1974.

....the order in which vertices, edges, or paths are processed, either subgraph is obtainable by the heuristic, thus achieving the desired worst case size ratio. 3.1 The Path Embedding Heuristic. The path embedding heuristic [CNS] is based on the linear time planarity algorithm of Hopcroft and Tarjan [HT]. Using depth first search, an initial cycle is found in a graph G, deleted, and then embedded in the plane. The remainder of G is then decomposed into edge disjoint paths and an attempt is made to embed each path inside or outside the cycle. If all paths can be embedded, the graph is planar; ....

....continues. We discuss AP in more detail. A depth first search (dfs) of G imposes both a vertex numbering and an edge orientation, yielding a digraph D = V; E) where E is partitioned into a set E T of tree edges and a set EF of fronds. Paths are recursively generated by procedure PATHFINDER (see [HT]) In order to avoid early wrong choices for embedding paths inside or outside the cycle C, one must generate paths systematically, choose appropriate regions for embedding them, and perhaps rearrange paths to accommodate new ones. Thus, after the initial dfs of G, the vertices and adjacency lists ....

[Article contains additional citation context not shown here]

J. Hopcroft and R.E. Tarjan. Efficient planarity testing. J. ACM 21 (4) (1974) 549-568.

....in which vertices, edges, or paths are processed, either subgraph is obtainable by the heuristic, thus achieving the desired worst case solution ratio. 3. 1 The Path Embedding Heuristic The path embedding heuristic of Chiba et al. 5] is based on the planarity algorithm of Hopcroft and Tarjan [10]. Using depth first search, an initial cycle is found in a graph G, deleted, and then embedded in the plane. The remainder of G is then decomposed into edge disjoint paths and an attempt is made to embed each path inside or outside the cycle. If all paths can be embedded then the graph is planar; ....

....search (dfs) of G imposes both a numbering on the vertices and an orientation on the edges, converting G into a directed graph D = V; E) such that E is partitioned into a set E T of tree edges and a set E F of fronds. Paths are recursively generated by the procedure PATHFINDER described in [10]. In order to avoid early wrong choices for embedding paths inside or outside the initial cycle C, it is necessary to generate paths systematically, to choose appropriate regions for embedding them, and, perhaps, to rearrange already embedded 2 paths to accommodate new ones. Thus, after the ....

[Article contains additional citation context not shown here]

J. Hopcroft and R.E. Tarjan, Efficient planarity testing, J. ACM, vol. 21, pp. 549-568, 1974.

....Figure 2 is devoted to Planarity. This is a critical issue in graph drawing, because planarity of a graph may be an important constraint imposed by practical applications (such as graphs representing printed circuit boards) The complexity for testing planarity for undirected graphs can be linear[67] (see Chapter 3.3 in Battista et al. 5] See also Mehlhorn and Mutzel[88] for a discussion on implementation issues) However, many applications impose the additional requirement that edges are all in the same direction (planar drawings often make use of edges going around some nodes to avoid ....

J. Hopcroft and R. E. Tarjan, "Efficient Planarity Testing", Journal of the ACM, Vol. 21 No. 4, pp. 549--568, 1974.

....can repeat infinitely the graph G 0 [G 1 (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 ....

J. Hopcroft and R. Tarjan, Efficient planarity testing, Jour. of the ACM 21(4) (1976) 549--568.

....Note that it might be that G is non planar as a labeled graph, even though the underlying G is planar. This is illustrated in Figure 3. We investigate: The Labeled Planarity Problem: When is a labeled graph planar There is a well known algorithm for determining if an unlabeled graph is planar [8]. This algorithm runs in time bounded by a linear function in the size of the graph (see [4] or [5] for a discussion of the analysis of algorithms) 7 However, an unlabeled graph can have many different planar embeddings. For example, consider the n bond which has a pair of vertices and n ....

J.E. Hopcroft and R.E. Tarjan, Efficient planarity testing, Jour. ACM 21(1974)549-568

....eventually adding dummy vertices for crossings. The planarization step is then followed by a representation step, that computes a drawing with the given embedding and removes the dummy vertices. Planarization Steps are implemented by using variants of well known planarization algorithms (see, e.g. [14]) Notice that the choice of a planar embedding can deeply affect the output of the drawing algorithm. For example, the well known algorithm in [20] that computes orthogonal drawings with the minimum number of bend can give rise to representations of a same graph that, depending on the choice of ....

J. Hopcroft and R. E. Tarjan. Efficient planarity testing. J. ACM, 21(4):549--568, 1974.

....according to the results in the last section. For the sake of completeness, let us show an algorithm to check the En such that G 2 En , for a given graph G. This algorithm uses a planarity algorithm in linear time. There are some algorithms for testing planarity in linear time (see, for instance, [11]) 8 E 2 E 1 E 0 E 3 E 4 E 5 E 6 E 7 E 8 E 9 E 10 E 16 E 15 E 14 E 13 E 12 E 11 6 6 6 6 L,M,T L,M,T L M,T L M,T L M,T L,M T L T M L,M,T oe L M,T oe L M,T oe L M,T L M,T 6 L M,T L,M,T 6 L M Phi Phi Phi Phi T 6 L M T 6 L 6 ....

....planar then G 2 E 6 . END. b) Check the planarity of T (L(G) If T (L(G) is not planar then G 2 E 8 . END. else G 2 E 9 . END. Notice that the algorithm works in an O(n) time, where n is the number of vertices of G. We have already remarked that testing planarity has a linear time complexity [11]. To check the planarity of L(G) G must be planar. If G is a graph with n vertices and m edges, then m 3n Gamma 6. So we can test the planarity of L(G) in an O(n) time. Testing L 2 (G) L 3 (G) M (G) T (G) M (L(G) and T (L(G) can be done in an O(n) time by a reasoning similar to the ....

J. E. Hopcroft and R. E. Tarjan. Efficient planarity testing. J. Assoc. Comput. March., 21:549-568, 1974.

....agreed to consider complexity classes as classes of languages over f0; 1g. However, now we not only want to talk about computations on classes of words (that is, languages) but also on classes of graphs or arbitrary relational structures. Example 8.1. A well known result of Hopcroft and Tarjan [44] says that PLANARITY of graphs is in PTIME. What this means is that there is a PTIME algorithm that, given any adjacency matrix of a graph, decides whether the graph is planar or not. In general, with each class C of structures we associate the language L(C) fe( A; A ) j A 2 C; A ....

J. E. Hopcroft and R. Tarjan. Efficient planarity testing. Journal of the ACM, pages 549--568, 1974.

....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 onto the plane. Hopcroft ....

....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 onto the plane. Hopcroft and Tarjan [28] showed that this can be done in O(n) time, where n is the number of vertices in the graph. The incremental planarity testing problem consists of performing an intermixed sequence of the following operations on a planar graph G with n vertices. Operation Insert(x; y) adds an edge (x; y) to G ....

[Article contains additional citation context not shown here]

J. Hopcroft and R. E. Tarjan. Efficient planarity testing. J. Assoc. Comput. Mach., 21:549--568, 1974.

....induced by Psi and adds to G a new vertex v c for each induced circuit c and a new edge (v c ; w) for each vertex w of c. The resulting augmented graph G is planar if and only if the ordering Psi is planar. Thus, the planarity of Psi can checked by running a planarity testing algorithm (e.g. [13]) on G . Besides its theoretical interest, however, this algorithm may be not be the most suited for practical applications, since the implementation of a linear time planarity testing algorithm is complex and requires sophisticated data structures. We show that there exists a much simpler ....

J. Hopcroft and R. E. Tarjan. Efficient planarity testing. J. ACM, 21(4):549--568, 1974.

....of graphs for which graph isomorphism is polynomial, and where the test of membership in F is also polynomial. We restrict the operations so that if the input pomset is not in F , then the result of the operation is the empty pomset. The class of planar graphs satisfies these two requirements [HT74, HW74] We assume in the following that the operations are restricted to pomsets whose (transitively reduced) graph representation is planar. This guarantees tractability of the operations. Such a restriction might seem artificial, but it is not unusual in computer science. In robotics for ....

J.E. Hopcroft and R.E. Tarjan. Efficient planarity testing. J. of the ACM, 21:549-- 568, 1974.

....complete for FP #P . Proof. This is the same as the proof of Corollary 5, except that the span P computation rejects any subgraph H which is not a tree. A more direct proof could be obtained by using a polynomial time canonical labelling algorithm for trees such as the one by Hopcroft and Tarjan [6]. ....

J. E. Hopcroft and R. E. Tarjan, Efficient planarity testing, Journal of the ACM 21 (1974) 549--568.

.... 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, ....

J. Hopcroft and R. E. Tarjan. Efficient planarity testing. J. ACM, 21(4):549--568, 1974.

....results for outerplanar graphs can be extended to series parallel graphs. Most of our algorithms do not use an embedding of the input graph. For sequential algorithms this leads to simpler algorithms. More importantly, the best known parallel algorithm for planar embedding takes time O(log 2 n) [14], so the use of an embedding would considerably slow down our parallel algorithms. These results give immediately algorithms for constructing compacted adjacency matrices: given a d bounded orientation of G, it is sufficient to store, for each v, only these neighbours x of v such that (v; x) ....

J.E. Hopcroft and R.E. Tarjan, Efficient planarity testing, J. Assoc. Comput. Mach. 21 (1974), pp. 549--568.

....time algorithm for obtaining the 3 connected components of a 2connected graph was devised by Hopcroft and Tarjan [8] There are well known linear time algorithms that for a given graph determine whether the graph is planar or not. The first such algorithm was obtained 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 ....

J. E. Hopcroft, R. E. Tarjan, Efficient planarity testing, J. ACM 21 (1974) 549--568.

....of attachment on a single branch of K, it is said to be local. local This paper is part of a larger project [JMM1, JMM2] which shows that there is a linear time algorithm to construct embeddings of graphs in an arbitrary (fixed) surface, generalizing the well known Hopcroft Tarjan algorithm [HT] for testing planarity in linear time. Our algorithms rely on the theory of bridges: a subgraph K of G is embedded in the surface and Supported in part by the Ministry of Science and Technology of Slovenia, Research Project P1 0210 101 94. 1 then this embedding is either extended to an ....

J. E. Hopcroft, R. E. Tarjan, Efficient planarity testing, J. ACM 21 (1974) 549--568.

....a graph G is a realization of G by the skeleton of a 3D convex polytope (see Fig. 2. The well known Steinitz s theorem says that a graph admits a 3D convex drawing if and only if it is planar and triconnected [80] see also Grunbaum [42] properties that can be verified in linear time (see, e.g. [48, 49]) Interestingly, it is a simple exercise to derive from the published proofs of Steinitz s theorem a cubic time method for constructing 3D convex drawings in the real RAM model [71] Unfortunately, this approach seems to require at least exponential volume and an exponential number of bits to ....

J. Hopcroft and R. E. Tarjan. Efficient planarity testing. J. ACM, 21(4):549--568, 1974.

....important among these are the terms: bridge, residual path, cross cut and ambitus . The definitions are similar to those used in the context of Tutte s Theorem on Hamiltonian circuits in 4 connected planar graphs given in Tutte [21] those in the planaritytesting algorithm of Hopcroft and Tarjan [8] or those in connection with the four color problem as presented in Ore [16] 4 Bidirectional Edges Problem: I x2.1 Graph Theoretic Terminology A graph G = V; E) is a finite set V of vertices and a set E of pairs of vertices, called edges . Either the edges are ordered pairs hu; vi of distinct ....

J. Hopcroft and R. Tarjan. Efficient Planarity Testing. Journal of Association for Computing Machinery, 21, October 1974. Acknowledgement 31

....corollary, we obtain that the thickness problem in the class of graphs without K 5 minors is solvable in linear time. Corollary 3.4 The thickness of a graph G without K 5 minors can be determined in linear time in the number of nodes of G. Proof. Apply a linear time planarity testing algorithm [HT74] to G. If G is planar, then (G) 1, otherwise (G) 2. 2 4 Other Invariants One may think that applying certain sum operations might also be applicable to control other topological invariants of graphs, such as the crossing number (G) or the skewness (G) of a graph G. The crossing number (G) ....

Hopcroft, J., and R.E. Tarjan, Efficient planarity testing, J. ACM 21 (1974), 549--568.

....by Psi and adds to G a new vertex v c for each induced circuit c and a new edge (v c ; w) for each vertex w of c. The resulting augmented graph G is planar if and only if the ordering Psi is planar. Thus, the planarity of Psi can be checked by running a planarity testing algorithm (e.g. [13]) on G . Besides its theoretical interest, however, this algorithm may be not be the most suited for practical applications, since the implementation of a linear time planarity testing algorithm is complex and requires sophisticated data structures. We show that there exists a much simpler ....

J. Hopcroft and R. E. Tarjan. Efficient planarity testing. J. ACM, 21(4):549-- 568, 1974.

....induced by Psi and adds to G a new vertex v c for each induced circuit c and a new edge (v c ; w) for each vertex w of c. The resulting augmented graph G is planar if and only if the ordering Psi is planar. Thus, the planarity of Psi can checked by running a planarity testing algorithm (e.g. [13]) on G . Besides its theoretical interest, however, this algorithm may not be the most suited for practical applications, since the implementation of a linear time planarity testing algorithm is complex and requires sophisticated data structures. We show that there exists a much simpler solution ....

J. Hopcroft and R. E. Tarjan. Efficient planarity testing. J. ACM, 21(4):549--568, 1974.

....restriction on the structure of a graph, some problems can be solved efficiently for planar graphs even if they are intractable in the general case. After detecting nonplanarity of a graph, using a standard planarity testing algorithm (see, e.g. Booth and Lueker [BL76] or Hopcroft and Tarjan [HT74]) it is often favorable to first extract a possible large planar subgraph and treat the remaining edges independently. If no edge can be added to this subgraph without destroying planarity, the subgraph is called a maximal planar subgraph. A planar subgraph of greatest cardinality is called a ....

Hopcroft, J., and R.E. Tarjan, Efficient planarity testing, J. ACM 21 (1974), 549--568.

....National Science Foundation grant CCR 9002428. 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. ....

....an O (mlogn) time solution to problem 2. Here m is the number of edges and n is the number of vertices of the input graph. On the other hand, the DFS tree approach was used only for problems 1 and 2: a linear time DFS tree algorithm (the HT algorithm) for problem 1 was given by Hopcroft and Tarjan [8] in 1974, and an O (mlogn) time algorithm for problem 2 was given by Cai, Han, and Tarjan [4] recently. The HT algorithm can also be extended to solve Problem 3, but the modification is complicated. The previous solutions for the four planar graph problems all consider biconnected graphs only. ....

[Article contains additional citation context not shown here]

Hopcroft, J. and Tarjan, R., "Efficient Planarity Testing," JACM, vol. 21, no. 4, pp. 549-568, October, 1974.

....multiple edges has at most 3n Gamma 6 edges. Thus in discussing time complexities for algorithms, linear time should be interpreted as time in O(n) There are several linear time algorithms for finding a planar embedding of a graph. The first of these was developed in 1974 by Hopcroft and Tarjan [20]. Booth and Lueker derived another approach which uses a data structure called a PQ tree [6] Other algorithms include those of de Fraysseix and Rosentiehl [10] and Williamson [34,35] More recently, Boyer and Myrvold have proposed a new approach [7] All of these algorithms are fairly complex. ....

J. Hopcroft and R. Tarjan. Efficient planarity testing. J. ACM, 21(4):549--568, 1974.

....Lemma 4.1 A planar y monotone LL drawing of a planar partition G = A [ B;E) can exist only if ffl the graph G LL defined above is planar, and ffl there exists a planar embedding of G LL such that any triangle containing vA or vB is a face. Planarity testing can be done in linear time [11]. Finding all triangles containing vA or vB can be done by marking the neighbors of vA and vB and then scanning all edges, this takes O(m) time. Testing whether a planar graph can be embedded without a separating triangle at all can be tested in linear time [2] However, in this context we need to ....

J. E. Hopcroft and R. E. Tarjan. Efficient planarity testing. Journal of the Association for Computing Machinery, 21(4):549--568, October 1974.

....operation, i) determine the two faces of G incident to a given edge, ii) determine the degree (number of adjacent edges) of any face or vertex, and (iii) traverse the edges incident to a given face or vertex in clockwise order. The noncrossing layout of G can be constructed in linear time using [HOP74], and the appropriate data structures can be produced in linear time using [WHI90] The Delta Wye Reduction Procedure consists of two phases, a labeling phase and a reduction phase. The labeling phase assigns to each edge and vertex of G a label indicating in a certain sense how far that edge or ....

Hopcroft, J. E. and R. E. Tarjan (1974). Efficient planarity testing, Journal of the ACM 21, 549--568.

....to compute a drawing with the minimal number of crossings [15] However, it is possible to test whether a graph Partially supported by DFG Grant Mu 1129 3 1, Forschungsschwerpunkt Effiziente Algorithmen fur diskrete Probleme und ihre Anwendungen . can be drawn without crossings in linear time [21]. In this paper, we restrict our attention to planar graphs. It is well known that planar n vertex graphs can be drawn without any bends on a grid of size (n Gamma 2) Theta (n Gamma 2) 9, 23, 8] However, the quality of these planar straight line drawings is not sufficient in practice. One ....

J. Hopcroft and R. E. Tarjan. Efficient planarity testing. J. ACM, 21(4):549--568, 1974.

....H 0 of G with edge set P and vertex set R. Hence jPj 3n Gamma 6, and H has less than 4n vertices. Since H is simple and bipartite, by Euler s formula it has at most 2(n jPj) Gamma 4 edges, which is less than 8n. 2 In particular, a map graph has a witness which may be checked in linear time [10], and so we have: Corollary 2.4 The recognition problem for map graphs is in NP. Let ff(G) denote the arboricity of a graph G, the minimum number of forests whose union is G. The next result is useful for the time analysis in Section 8. Corollary 2.5 A k map graph with n vertices has O(kn) ....

....an efficient algorithm to decide whether a given G has an atlas; it will either return an atlas or report failure. Furthermore, the algorithm returns a well formed atlas whenever possible (see Corollary 4.2) If G has an atlas, then by Lemma 2. 3 it has a witness graph checkable in linear time [10]. So in fact we describe an algorithm that makes the following assumption: Assumption 1 G has an atlas. If G does not have an atlas, we will discover this when our algorithm either fails, returns an invalid atlas, or takes more time than allowed by our analysis in Section 8. Since we assume that ....

[Article contains additional citation context not shown here]

J. Hopcroft and R. E. Tarjan. Efficient planarity testing. J. ACM, 21(4):549--568, 1974.

....edges. For each cycle C 2 C the triangular faces of GC were faces in CA(G) so C is now the boundary of a face. Thus, we found a face boundary constrained embedding of G, and Lemma 3.1 is proved. 2 Since jCj 2n Gamma 4, we add m jCj = O(n) new vertices. Planarity can be tested in linear time [15], so we can test the existence of a face boundary constrained embedding for biconnected graphs in linear time. 3.2 Connected graphs To test the existence of a face boundary constrained embedding for a connected graph G, we proceed by induction on the number of cut vertices of G. If there are ....

J. E. Hopcroft and R. E. Tarjan. Efficient planarity testing. Journal of the Association for Computing Machinery, 21(4):549--568, October 1974.

....algorithm using PQ trees, a very complicated data structure. It was proved to have a linear time implementation in 1976 partly by Even and Tarjan [ET] and partly by Booth and Lueker [BL] by introducing an st numbering of a graph. The first linear time algorithm was given by Hopcroft and Tarjan [HT] in 1974 using depth first search trees. This algorithm, which is path addition based, is generally considered to be more complicated than the PQ tree approach. Many applications require not only testing planarity but also embedding a planar graph in the plane. The embedding problem for a ....

Hopcroft, J. and R. Tarjan, Efficient Planarity Testing, JACM, vol. 21, No. 4, pp. 549-568, 1974.

....of possible planar embeddings of G in the plane in case G is planar. Linear time algorithms for P1, P3, and P4 have been known for a long time. The first linear time solution (which we call the H T algorithm) for problem P1 ( the planarity testing problem) was given by Hopcroft and Tarjan [7] in 1974 using depth first search (DFS) trees. A P Q tree solution for P1 based on an earlier algorithm given by Lempel, Even, and Cederbaum [11] was proved to have a linear time implementation in 1976 partly by Even and Tarjan [4] and partly by Booth and Lueker [1] The P Q tree approach is ....

....each tree edge. Also, the H T algorithm processes one path at a time, while our algorithm processes one edge at a time. In this sense, our algorithm is a more recursive version of the H T algorithm. For the above reason, many of our lemmas and theorems are similar, but not identical, to those in [7]. Instead of referring the readers to [7] for the proofs, we find it more convenient and accurate to supply all main proofs in this paper. The rest of this paper is organized as follows. Section 2 gives preliminary definitions. Section 3 is a new version of the H T planarity testing algorithm, ....

[Article contains additional citation context not shown here]

Hopcroft, J. and Tarjan, R., "Efficient Planarity Testing," JACM, vol. 21, no. 4, pp. 549-568, October, 1974.

No context found.

J. Hopcroft and R.E. Tarjan, Efficient planarity testing, J. ACM 21 (1974), 549--568.

No context found.

J. Hopcroft and R. Tarjan, "Efficient Planarity Testing," Journal of the ACM, vol. 21, no. 4, pp. 549-568, 1974.

No context found.

John Hopcroft and Robert E. Tarjan. Efficient planarity testing. Journal of the ACM, 21(4):549--568, 1974.

No context found.

J. Hopcroft and R. E. Tarjan. Efficient planarity testing. J. ACM, 21(4):549--568, 1974.

No context found.

John Hopcroft and Robert Tarjan. Efficient planarity testing. JACM, 21(4):549--568, 1974.

No context found.

J. E. Hopcroft and R. E. Tarjan. Efficient planarity testing. Journal of the ACM, 21, (1974), 549--568.

First 50 documents  Next 50

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC