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RealTime Minimum Vertex Cover For TwoTerminal SeriesParallel Graphs
 Proceedings of the Thirteenth Conference on Parallel and Distributed Computing and Systems
, 2000
"... Tree contraction is a powerful technique for solving a large number of graph problems on families of recursively definable graphs. The method is based on processing the parse tree associated with a member of such a family of graphs in a bottomup fashion, such that the solution to the problem is ..."
Abstract

Cited by 8 (8 self)
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Tree contraction is a powerful technique for solving a large number of graph problems on families of recursively definable graphs. The method is based on processing the parse tree associated with a member of such a family of graphs in a bottomup fashion, such that the solution to the problem is obtained at the root of the tree. Sequentially, this can be done in linear time with respect to the size of the input graph. In parallel, efficient and even cost optimal tree contraction algorithms have also been developed. In this paper we show how the method can be applied to compute the cardinality of the minimum vertex cover of a twoterminal seriesparallel graph. We then construct a realtime paradigm for this problem and show that in the new computational environment, a parallel algorithm is superior to the best possible sequential algorithm, in terms of the accuracy of the solution computed. Specifically, there are cases in which the solution produced by a parallel algorithm ...
Parallel Algorithms for Hamiltonian Problems on Quasithreshold Graphs
 Parallel and Distributed Computing
, 1998
"... In this paper we show structural and algorithmic properties on the class of quasithreshold graphs, or QTgraphs for short, and prove necessary and sufficient conditions for a QTgraph to be Hamiltonian. Based on these properties and conditions, we construct an efficient parallel algorithm for findi ..."
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Cited by 7 (7 self)
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In this paper we show structural and algorithmic properties on the class of quasithreshold graphs, or QTgraphs for short, and prove necessary and sufficient conditions for a QTgraph to be Hamiltonian. Based on these properties and conditions, we construct an efficient parallel algorithm for finding a Hamiltonian cycle in a QTgraph; for an input graph on n vertices and m edges, our algorithm takes O(log n) time and requires O(n + m) processors on the CREW PRAM model. In addition, we show that the problem of recognizing whether a QTgraph is a Hamiltonian graph and the problem of computing the Hamiltonian completion number of a non Hamiltonian QTgraph can also be solved in O(log n) time with O(n + m) processors. Our algorithms rely on O(log n)time parallel algorithms, which we develop here, for constructing tree representations of a QTgraph; we show that a QTgraph G has a unique tree representation, that is, a tree structure which meets the structural properties of G. We also present parallel algorithms for other optimization problems on QTgraphs which run in O(log n) time using a linear number of processors.
A Fast Parallel Algorithm to Recognize P 4 sparse Graphs
 Discrete Appl. Math
"... A number of problems in computational semantics, groupbased collaboration, automated theorem proving, networking, scheduling, and cluster analysis suggested the study of graphs featuring certain "local density" characteristics. Typically, the notion of local density is equated with the absence of c ..."
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Cited by 3 (1 self)
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A number of problems in computational semantics, groupbased collaboration, automated theorem proving, networking, scheduling, and cluster analysis suggested the study of graphs featuring certain "local density" characteristics. Typically, the notion of local density is equated with the absence of chordless paths of length three or more. Recently, a new metric for local density has been proposed, allowing a number of such induced paths to occur. More precisely, a graph G is called P4sparse if no set of five vertices in G induces more than one chordless path of length three. P4sparse graphs generalize the wellknown class of cographs corresponding to a more stringent local density metric. One remarkable feature of P4sparse graphs is that they admit a tree representation unique up to isomorphism. In this work we present a parallel algorithm to recognize P4sparse graphs and show how the data structures returned by the recognition algorithm can be used to construct the corresponding tr...