S. Even and R.E. Tarjan, Computing an st-numbering,T6)6C4w Comput. Sci., 2 (1976), pp. 339--344.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....other vertex v i , 1 i n, with deg(v i ) 2, we have pred(v i ) 1 and succ(v i ) 1. Lempel et al. 22] show that for any biconnected graph G = V; E) and for any s; t 2 V , there exists an st ordering of G. Cheriyan and Reif [6] extended this result to directed graphs. Even and Tarjan [17] develop a linear time algorithm to compute an st ordering of an undirected biconnected graph (also see [16, 24, 29] Under the guise of bipolar orientations, st orderings have also been studied in [7, 10, 26] In related work, Papakostas and Tollis [25] describe an algorithm for producing ....

S. EVEN AND R. E. TARJAN, Computing an st-numbering. Theoret. Comput. Sci., 2(3):339--344, 1976.

....the number of bends is at most 3m n 2, and by Theorem 2 the volume is at most (n (m n 2) 3) 2n m 2) 3) If G is 6 regular then m = 3n and the average number of bends per edge is at most 2 3 o(1) and the volume is O n 2 . Using the algorithm of Even and Tarjan [12] and by Theorem 2, the st ordering and the drawing itself can be determined in O(n) time. Note that for a non biconnected graph G with a constant number of end blocks, a similar method to the above (see [28] establishes the same upper bounds on the number of bends and the volume of a drawing of ....

S. Even and R. E. Tarjan, Computing an st-numbering, Theoret. Comput. Sci., 2(1976), 339-344.

....combinatorial structures that have been successfully applied to solving various graph problems. Here, we overview previous 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 ....

....33] of G witit respect to an edge (s,t) is an orientation of the edges of G such that the resulting digraph D is acyclic, s is the unique source of D, and t is the unique sink of D. A biconnected graph admits a bipolar orientation with respect to any edge (s, which can be computed in linear time [18]. An st numbering of G is a numbering v, v of the vertices of G such that s = v, t = v, and each other vertex vi, 1 i n, is adjacent to at least one vertex vj, j i, and to at least one vertex vk, k i. Given a bipolar orientation of a biconnected graph G, we construct two spanning trees of ....

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S. Even and R. E. Tarjan. Computing an st-numbering. Theoret. Cornput. Sci., 2:339 344, 1976.

....for every other vertex vi, i i n, with deg(vi) 2, we have pred(vi) i and succ(vi) 1. Lempel et al. 22] show that for any biconnected graph G = V, E) and for any s, t V, there exists an st ordering of G. Cheriyan and Reif [6] extended this result to directed graphs. Even and Tarjan [16] develop a linear time algorithm to compute an st ordering of an undirected biconnected graph. Under the guise of bipolar orientations, st orderings have also been studied in [7, 10, 25] For graphs of maximum degree four, Papakostas and Tollis [24] give an algorithm for producing a so called bst ....

....numbering of the blocks. Hence si has predecessors in Bj, and therefore the only vertex with zero predecessors is s. 10 The block cut tree can be determined in O(n m) time. If the block Bi has ni vertices and mi edges, then the st ordering of block Bi can be determined in O(ni mi) time [16]. Hence the overall time is O(i(ni mi) O(m i hi) Only the cut vertices are double counted in this sum, and each time a cut vertex is counted can be attributed to a unique endpoint of an edge, hence i ni = O(n m) and the result follows. Theorem 3. Given a non biconnected n vertex m edge ....

S. EVEN AND R. E. TARJAN, Computing an st-numbering. Theoret. Cornput. Sci., 2(3):339-344, 1976.

....experimentation algorithms such as those presented in [3, 17, 22] which deal with planar and or triconnected graphs only. For what concerns the output, it is accepted to be non plane, even if the input graph is planar. The chosen algorithms add one vertex at a time according to an st numbering [13]; therefore, the algorithm described in [20] has not been considered because it works on pairs of vertices. All the algorithms first work on biconnected components and then splice the drawings of the components to form a drawing of the entire graph. Thus, the results of our experiments are ....

....edges. After drawing the resulting cubic Figure 3: Transformation from biconnected at most cubic graph to cubic graph and vice versa. graph, we can reinserr a vertex of degree 2 adjacent to vertices i and j by simply placing a dot along the drawing of edge (i, j) see Fig. 3) Definition 4 [13] Oive ay edge s,t of a bicomected graph G = V, E) a fuctio g : V 1,2, is called a st numbering if the followig coditios hold: g( # g(v) for # g) 1 From now on, we shall refer to the vertices of a biconnected st numbered cubic graph G: V, E) by their st numbers. 3 The ....

S. Even, and R.E. Tarjan. Computing an st-numbering. Theoret. Comp. Sci., 365 372, 1987.

....to ear decomposition is st numbering. This is an ordering of the vertices such that there is an edge between the first and last vertices s and t, and such that each other vertex has neighbors both before and after it in the ordering. An st numbering can be computed from an open ear decomposition [18]; Maon et al. 46] showed how to do this in parallel. Klein and Reif [38] used st numbering as part of a parallel algorithm for embedding planar graphs. Ear decompositions have also been used as part of other graph algorithms, in particular for the connectivity of a graph. The algorithms we ....

S. Even and R.E. Tarjan, Computing an st-numbering. Theor. Comput. Sci. 2, 1976, 339--344.

....in [29] entails that our solutions to edge failure are a superset than those offered by [29] A. Overview of the Previous Algorithm In order to compare our scheme to the one presented in [29] it is necessary to reiterate most of the algorithm in [29] The scheme in [29] determines an numbering ([20], 39] Fig. 9. Another example of B and R trees using the same node numbering. denoted by Let us select two nodes, and in such that and are end points of some edge in An numbering on the graph is defined to be such that and every vertex has two adjacent vertices, and which satisfy Note that an ....

S. Even and R. E. Tarjan, "Computing an st-numbering," Theoretical Comput. Sci., vol. 2, pp. 339--344, 1976.

....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 ....

....of G having s and t in the contour of the same face [38] A planar st orientation of an undirected graph G is an orientation of the edges of G yielding a planar st graph. It is known that every biconnected planar graph has an st orientation, and such an orientation can be found in O(n) time [15]. Note that each pertinent graph in T 3 (G) is biconnected because of Lemma 2.2. Consequently, each pertinent graph Gamma i in T 3 (G) has an st orientation. 7 Fact 2.2 [7] Let G = V; E) be a biconnected graph, with a given st orientation with source s and sink t. Let T 3 (G) be the tree of ....

S. Even and R. E. Tarjan. Computing an st-numbering. Theoret. Comput. Sci., 2:339--344, 1976.

.... 2, and by Theorem 2 the volume is at most (n (m n 2) 3) 3 = 2n m 2) 3) 3 . If G is 6 regular then m = 3n and the average number of bends per edge is at most 2 2 3 o(1) and the volume is ( 5 3 n) 3 O n 2 = 4:63n 3 o n 3 . Using the algorithm of Even and Tarjan [12] and by Theorem 2, the st ordering and the drawing itself can be determined in O(n) time. Note that for a non biconnected graph G with a constant number of end blocks, a similar method to the above (see [28] establishes the same upper bounds on the number of bends and the volume of a drawing of ....

S. Even and R. E. Tarjan, Computing an st-numbering, Theoret. Comput. Sci., 2(1976), 339-344.

.... 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, ....

S. Even and R. E. Tarjan. Computing an st-numbering. Theoret. Comput. Sci., 2:339--344, 1976.

....primal graph G and output the set of nontrivial sccs end. The correctness of Algorithm 2 follows from the informal discussion above. The details of the proof are omitted. 7 CONCLUSION AND OPEN PROBLEMS 22 Step 1 can be implemented in linear time by various algorithms presented in ( 5] 7] [3], 1] 9] Steps 2 and 13 can be implemented in linear time. Steps 4, 5 and 6 can be implemented in linear time. Since the number of nontrivial sccs in G is greater than or equal to p n at the beginning of each iteration of the do loop in step 3 of Algorithm 2, at least p n edges are ....

Even, S. and Tarjan, R. "Computing an st-numbering" Theoretical Computer Science, vol. 2, 1976, p339-344.

....any edge crossings. A graph is obviously planar, if and only if its biconnected components are planar. We therefore assume that G is biconnected. The planarity testing algorithm of Lempel, Even, and Cederbaum (1967) first labels the vertices of G as 1; 2: n using an st numbering (see Even and Tarjan, 1976). The st numbering induces an orientation of the graph, in which every edge is directed from the incident vertex with the higher st number towards the incident vertex with the lower st number. From now on we refer to the vertices of G by their st numbers and call an edge (v; u) with u v, ....

Even, S. and Tarjan, R. E. (1976). Computing an st-numbering. Theoretical Computer Science, 2, 339--344.

....combinatorial structures that have been successfully applied to solving various graph problems. Here, we overview previous 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 ....

....of G with respect to an edge (s; t) is an orientation of the edges of G such that the resulting digraph D is acyclic, s is the unique source of D, and t is the unique sink of D. A biconnected graph admits a bipolar orientation with respect to any edge (s; t) which can be computed in linear time [18]. An st numbering of G is a numbering v 1 ; v n of the vertices of G such that s = v 1 , t = v n , and each other vertex v i ; 1 i n, is adjacent to at least one vertex v j ; j i, and to at least one vertex v k ; k i. Given a bipolar orientation of a biconnected graph G, we ....

[Article contains additional citation context not shown here]

S. Even and R. E. Tarjan. Computing an st-numbering. Theoret. Comput. Sci., 2:339--344, 1976.

....different linear time algorithms for the problem of testing planarity of a graph. Lempel, Even and Cederbaum [LEC] presented the vertex addition based 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 ....

Even, S. and R.E. Tarjan, Computing an st-numbering, Th. Comp. Sci., vol. 2, pp. 339-344, 1976.

....( 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 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 ....

....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 12 13 14 3. ....

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Even, S. and Tarjan, R. E., "Computing an st-numbering," Th. Comp. Sci., vol. 2, no. 3, pp. 339-344, 1976.

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S. Even and R.E. Tarjan, Computing an st-numbering,T6)6C4w Comput. Sci., 2 (1976), pp. 339--344.

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S. Even and R. Tarjan, Computing an st-numbering, Theoret. Comput. Sci., 2 (1976), pp. 339--344.

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S. Even and R. E. Tarjan, Computing an st-numbering. Theoret. Comput. Sci., 2:339-344, 1976.

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S. Even and R. E. Tarjan, Computing an st-numbering. Theoret. Comput. Sci., 2:339-344, 1976.

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S. Even and R. E. Tarjan, Computing an st-numbering. Theoret. Comput. Sci., 2:339-344, 1976.

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S. Even and R. E. Tarjan. Computing an stnumbering. Theoret. Comput. Sci., 2:339-344, 1976.

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S. Even and R. E. Tarjan, Computing an st-numbering. Theoret. Comput. Sci., 2(3):339-344, 1976.

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S. Even and R. E. Tarjan, "Computing an st-numbering," Theoretical Computer Science 2 (1976), 339-344.

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S. Even and R. E. Tarjan, Computing an st-numbering. Theoret. Comput. Sci., 2(3):339--344, 1976.

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S. Even and R. E. Tarjan, Computing an st-numbering. Theoret. Comput. Sci., 2:339-344, 1976.

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