A. Kanevsky and V. Ramachandran. Improved algorithms for graph four-connectivity. J. Comp. Syst. Sc., 42:288--306, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... open ear decomposition if and only if it is biconnected [46] Lovasz showed that the problem of computing an open ear decomposition in parallel is in NC [30] Ear decomposition has also been used in designing efficient sequential and parallel algorithms for triconnectivity [37] and 4 connectivity [23]. In addition to graph connectivity, ear decomposition has been used in graph embeddings (see [9] The sequential algorithm: Ramachandran [37] gave a linear time algorithm for ear decomposition based on depth first search. Another sequential algorithm that lends itself to parallelization (see ....

A. Kanevsky and V. Ramachandran. Improved Algorithms for Graph Four-Connectivity. Journal of Computer and System Sciences, 42(3):288--306, 1991. Bader, Illendula, Moret, and Weisse-Bernstein

.... log r)jV (G)j edges [Tho01] This implies that for any single crossing minor free graph G, jE(G)j = O(jV (G)j) To run the algorithm, we apply algorithms to obtain all connected components and 1 cuts in linear time [Tar72] all 2 cuts [HT73,MR92] and all O(n ) 3 cuts in O(n time [KR91]. Checking whether a particular 3 cut is strong can be accomplished in O(n) time using depth rst search. All other operations, including checking if a graph is planar or has treewidth at most c H , can be performed in linear time [Wil84] To set up recurrence relations, we make the assumption ....

Arkady Kanevsky and Vijaya Ramachandran. Improved algorithms for graph four-connectivity. J. Comput. System Sci., 42(3):288-306, 1991.

.... number of k sums and k 3) In linear time we can obtain all 1 cuts [Tar72] and we can obtain all 2 cuts using the algorithms of Hopcroft and Tarjan [HT73] or Miller and Ramachandran [MR92] The number of 3 cuts in a 3connected graph is O(n ) and we can obtain all 3 cuts in O(n ) time [KR91]. We can check whether each 3 cut is strong in O(n) time using a depth rst search. All other operations including checking planarity and having treewidth at most c H can be performed in linear time [Wil84,Bod96] Now, if the algorithm makes no recursive calls, the running time of the algorithm, ....

Arkady Kanevsky and Vijaya Ramachandran. Improved algorithms for graph four-connectivity. J. Comput. System Sci., 42(3):288-306, 1991. Twenty-Eighth IEEE Symposium on Foundations of Computer Science (Los Angeles, CA, 1987).

.... or biconnected, although in the latter case a simpler algorithm was known [66] Miller and Ramachandran [51] used ear decomposition as part of an algorithm for testing graph 3 vertex connectivity; this algorithm has recently been improved by Ramachandran and Vishkin [56] Kanevsky and Ramachandran [34] again used ear decomposition to test 4 vertex connectivity. Eppstein [17] has recently found another application of open ear decomposition, to the recognition and decomposition of series parallel graphs. His algorithm runs in time O(log n) with linear CRCW processors, improving a previous ....

A. Kanevsky and V. Ramachandran, Improved Algorithms for Graph Four-Connectivity. 28th Symp. Found. Comput. Sci., 1987, 252--259.

.... not triconnected (i.e. 3 vertex connected) is also well known and is represented as a 3 block graph [HT73, Tut66] The 3 block graph is also extended for a non biconnected graph [DBT90, HR91] Recently, the structure of a graph that is not four connected (i.e. 4vertex connected) is studied in [Hsu92, KR91, KTDBC91] with the emphasis on triconnected graphs. Several properties of a (k Gamma 1) vertex connected graph that is not k vertex connected are also studied in [CBKT93, Mat72, Mat78] In this paper, we study the structure of an undirected graph that is not four connected detailing on the parts we need ....

....to both B 1 and B 2 and whose cardinality is less than . Furthermore, B 1 B 2 is in every separating set which separates a vertex in B 1 n B 2 and a vertex in B 2 n B 1 . Proof: This claim follows from the structural descriptions of all 2 blocks [Har69] 3 blocks [HT73, Tut66] and 4 blocks [Hsu92, KR91, KTDBC91]. 2 Claim 4.5 Let B 1 and B 2 be two blocks, 4. Let I = B 1 B 2 . For reaching k vertex connectivity, k 4, there is no special block in I. Proof: If 1, then I = Thus we assume that 1. We also know that if = 4, then there are at least four internally vertex disjoint paths ....

A. Kanevsky and V. Ramachandran. Improved algorithms for graph fourconnectivity. J. Comp. System Sci., 42:288--306, 1991.

....Hence we are unable to obtain efficient parallel algorithms by parallelizing sequential algorithms based on depth first search or breadth first search. Instead, an alternative search technique called ear decomposition has proved to be a very useful tool for designing parallel graph algorithms [KR91a, KR90, MSV86, MR92, Ram93, RR89] Combined with an efficient parallel routine for finding connected components [AS87] and the Euler tour technique [TV85] this gives efficient parallel algorithms for several important problems on undirected graphs which include various connectivity problems ....

.... KR90, MSV86, MR92, Ram93, RR89] Combined with an efficient parallel routine for finding connected components [AS87] and the Euler tour technique [TV85] this gives efficient parallel algorithms for several important problems on undirected graphs which include various connectivity problems [KR91a, FRT93, MR92, Ram93] st numbering [MSV86] planarity testing and embedding [RR89] finding a strong orientation and finding a minimum cost spanning forest. Figure 10.2 illustrates the building blocks for designing parallel graph algorithms using ear decomposition, the Euler tour technique and ....

A. Kanevsky and V. Ramachandran. Improved algorithms for graph four-connectivity. J. Comp. System Sci., 42:288--306, 1991.

....to find triconnected components. Miller and Ramachandran [31] have recently proposed a parallel algorithm to identify triconnected components in O(log 2 n) time with O(m) processors. This places triconnectivity in NC. Four connected components are more difficult, but Kanevsky and Ramachandran [25] have recently found an O(n 2 ) time algorithm. They also discovered a parallel implementation of their algorithm that runs in O(log n) time using O(n 2 ) processors. So the problem of partial reflections is in NC in both two and three dimensions. For k greater than 4, the question of ....

A. Kanevsky and V. Ramachandran, Improved algorithms for graph four-connectivity, in Proc. 28th IEEE Annual Symposium on Foundations of Computer Science, Los Angeles, October 1987, pp. 252--259.

....1 Introduction Graph connectivity has become a widely studied component of graph theory. To date, most of the work in this area has dealt with determining if an undirected graph contains at least two, three, or more vertex or edge disjoint paths between every pair of vertices (see, e.g. [5, 7, 8, 11, 12, 14, 15, 17, 20, 21]) or if a directed graph possesses analogous properties (see, e.g. 5, 16] This paper investigates a different sort of connectivity problem: determining if a graph has at most some number of distinct paths between every pair of vertices. In particular, we say a directed graph is singly ....

A. Kanevsky and V. Ramachandran. Improved algorithms for graph four-connectivity. In Proc. 28th IEEE Symp. on Foundations of Computer Science, pages 252--9, 1987.

....conditions derived in [15] for a graph to be uniquely realizable in d dimensions are: 1) vertex (d 1) connectivity, and (2) redundant rigidity. Of these, vertex connectivity is a well studied graph property and, efficient algorithms for verifying (d 1) connectivity have been developed [1, 5, 16, 18]. Redundant rigidity is less familiar, but efficient algorithms are known [15] and reviewed below. For later reference we will need to review some simple rigidity theory in x2.1 and x2.2; a more complete discussion can be found in some of the references [2, 3, 9, 28] In x2.3 a previously ....

....evidence that such graphs are uncommon if they exist at all. The heart of the procedure described in Fig. 2, is finding (d 1) vertex connected subgraphs and redundantly rigid subgraphs. The vertex connectivity problem is well studied, and good algorithms for finding maximal subgraphs are known [1, 5, 16, 18]. However, algorithms for finding redundantly rigid subgraphs have not been previously considered. In one dimension, this requires finding biconnected components, for which there are O(m) algorithms [1] In two dimensions, an O(n 2 ) algorithm for finding maximal redundantly rigid components is ....

A. Kanevsky and V. Ramachandran, Improved algorithms for graph four-connectivity, in Proc. 28th IEEE Annual Symposium on Foundations of Computer Science, Los Angeles, October 1987, pp. 252--259.

....last section we give some pointers towards obtaining optimal logarithmic time parallel algorithms for graph biconnectivity and triconnectivity. Open ear decomposition has been used to obtain efficient parallel algorithms for several other important graph problems such as graph four connectivity [KR91], st numbering [MSV86] and graph planarity [RR89] Algorithmic Notation The algorithmic notation in this report is from Tarjan [Ta83] We enclose comments between a pair of curly brackets with asterisks ( f and g ) We incorporate parallelism by use of the following statement that augments ....

A. Kanevsky, V. Ramachandran, "Improved algorithms for graph fourconnectivity, " Jour. Comput. Syst. Sci., vol. 42, 1991, pp. 288-306.

....Gabow has devised a very nice algorithm for edge connectivity. His algorithm, unlike previous algorithms for connectivity, does not appeal to Menger s theorem. It runs in O(km log(n 2 =m) time [9] The algorithms for vertex connectivity for 3 k p n currently require O(k 2 n 2 ) time [14], 2] 20] The subject of this paper is the parallel complexity of 3 vertex connectivity. The importance of 3 vertex connectivity stems from the fact that if a planar graph is 3vertex connected (triconnected) then it has a unique embedding on a sphere. Hence an efficient algorithm that divides ....

A. Kanevsky, V. Ramachandran, "Improved algorithms for graph four-connectivity," Journal of Computer and System Sciences, 42 (1991), pp. 288--306.

....[7] Hence we are unable to obtain efficient parallel algorithms by parallelizing sequential algorithms based on depth first search or breadth first search. Instead, an alternative search technique called ear decomposition has proved to be a very useful tool for designing parallel graph algorithms [7, 6, 10, 17, 20, 19]. Combined with an efficient parallel routine for finding connected components [1] and the Euler tour technique [24] we have efficient parallel algorithms for several important graph problems which include various connectivity problems [6, 17, 19] st numbering [10] planarity testing and ....

.... for designing parallel graph algorithms [7, 6, 10, 17, 20, 19] Combined with an efficient parallel routine for finding connected components [1] and the Euler tour technique [24] we have efficient parallel algorithms for several important graph problems which include various connectivity problems [6, 17, 19], st numbering [10] planarity testing and embedding [20] finding a strong orientation and finding a minimum cost spanning forest z . Figure 1 illustrates the building blocks for designing parallel graph algorithms using ear decomposition, the Euler tour technique and the routine for finding ....

A. Kanevsky and V. Ramachandran, Improved algorithms for graph four-connectivity, J. Comp. System Sci. 42 (1991), 288--306.

....[7] Hence we are unable to obtain efficient parallel algorithms by parallelizing sequential algorithms based on depth first search or breadth first search. Instead, an alternative search technique called ear decomposition has proved to be a very useful tool for designing parallel graph algorithms [7, 6, 10, 17, 20, 19]. Combined with an efficient parallel routine for finding connected components [1] and the Euler tour technique [24] we have efficient parallel algorithms for several important graph problems which include various connectivity problems [6, 17, 19] st numbering [10] planarity testing and ....

.... for designing parallel graph algorithms [7, 6, 10, 17, 20, 19] Combined with an efficient parallel routine for finding connected components [1] and the Euler tour technique [24] we have efficient parallel algorithms for several important graph problems which include various connectivity problems [6, 17, 19], st numbering [10] planarity testing and embedding [20] finding a strong orientation and finding a minimum cost spanning forest y . Figure 1 illustrates the building blocks for designing parallel graph algorithms using ear decomposition, the Euler tour technique and the routine for finding ....

A. Kanevsky and V. Ramachandran, Improved algorithms for graph four-connectivity, J. Comp. System Sci. 42 (1991), 288--306.

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A. Kanevsky and V. Ramachandran. Improved algorithms for graph four-connectivity. J. Comp. Syst. Sc., 42:288--306, 1991.

No context found.

Arkady Kanevsky and Vijaya Ramachandran. Improved algorithms for graph four-connectivity. J. Comput. System Sci., 42(3):288-306, 1991.

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A. Kanevsky and V. Ramachandran. Improved algorithms for graph four-connectivity. J. Comp. Syst. Sc., 42:288--306, 1991.

No context found.

Arkady Kanevsky and Vijaya Ramachandran. Improved algorithms for graph four-connectivity. J. Comput. System Sci., 42(3):288--306, 1991.

No context found.

Arkady Kanevsky and Vijaya Ramachandran. Improved algorithms for graph four-connectivity. J. Comput. System Sci., 42(3):288--306, 1991. Twenty-Eighth IEEE Symposium on Foundations of Computer Science (Los Angeles, CA, 1987).

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