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OutputSensitive Reporting of Disjoint Paths
, 1996
"... A kpath query on a graph consists of computing k vertexdisjoint paths between two given vertices of the graph, whenever they exist. In this paper, we study the problem of performing kpath queries, with k < 3, in a graph G with n vertices. We denote with the total length of the paths reported. For ..."
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Cited by 11 (2 self)
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A kpath query on a graph consists of computing k vertexdisjoint paths between two given vertices of the graph, whenever they exist. In this paper, we study the problem of performing kpath queries, with k < 3, in a graph G with n vertices. We denote with the total length of the paths reported. For k < 3, we present an optimal data structure for G that uses O(n) space and executes kpath queries in outputsensitive O() time. For triconnected planar graphs, our results make use of a new combinatorial structure that plays the same role as bipolar (st) orientations for biconnected planar graphs. This combinatorial structure also yields an alternative construction of convex grid drawings of triconnected planar graphs.
Optimal Independent Spanning Trees on Hypercubes
, 2004
"... Two spanning trees rooted at some vertex r in a graph G are said to be independent if for each vertex v of G, v ≠ r, the paths from r to v in two trees are vertexdisjoint. A set of spanning trees of G is said to be independent if they are pairwise independent. A set of independent spanning trees is ..."
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Cited by 3 (1 self)
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Two spanning trees rooted at some vertex r in a graph G are said to be independent if for each vertex v of G, v ≠ r, the paths from r to v in two trees are vertexdisjoint. A set of spanning trees of G is said to be independent if they are pairwise independent. A set of independent spanning trees is optimal if the average path length of the trees is the minimum. Any kdimensional hypercube has k independent spanning trees rooted at an arbitrary vertex. In this paper, an O(kn) time algorithm is proposed to construct k optimal independent spanning trees on a kdimensional hypercube, where n = 2 k is the number of vertices in a hypercube.
Independent Tree Spanners  FaultTolerant Spanning Trees with Distance Guarantees
 In Proceedings 24rd International Workshop on GraphTheoretic Concepts in Computer Science, WG'98
, 1998
"... . For any fixed rational parameter t 1, a (tree) tspanner of a graph G is a spanning subgraph (tree) T in G such that the distance between every pair of vertices in T is at most t times their distance in G. General tspanners and their variants have multiple applications in the field of communi ..."
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Cited by 2 (2 self)
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. For any fixed rational parameter t 1, a (tree) tspanner of a graph G is a spanning subgraph (tree) T in G such that the distance between every pair of vertices in T is at most t times their distance in G. General tspanners and their variants have multiple applications in the field of communication networks, distributed systems, and network design. In this paper, we combine the two concepts of simple structured, sparse tspanners and faulttolerance by examining independent tree tspanners. Given a root vertex r, this is a pair of tree tspanners, such that the two paths from any vertex to r are edge disjoint or internally vertex dijoint, respectively. It is shown that a pair of independent tree tspanners can be found in linear time for t ! 3, whereas the problem for arbitrary t 4 is NPcomplete. As a less restrictive concept, we also treat tree trootspanners, where the distance constraint is relaxed. Here, we show that the problem of finding an independent pair of su...
Independent Spanning Trees With Small Depths in Iterated Line Digraphs
, 1997
"... . We show that the independent spanning tree conjecture on digraphs is true if we restrict ourselves to line digraphs. Also, we construct independent spanning trees with small depths in iterated line digraphs. From the results, we can obtain independent spanning trees with small depths in de Bruijn ..."
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Cited by 2 (0 self)
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. We show that the independent spanning tree conjecture on digraphs is true if we restrict ourselves to line digraphs. Also, we construct independent spanning trees with small depths in iterated line digraphs. From the results, we can obtain independent spanning trees with small depths in de Bruijn and Kautz digraphs that improve the previously known upper bounds on the depths. Keywords. independent spanning trees, line digraphs, vertexconnectivity, de Bruijn digraphs, Kautz digraphs, interconnection networks, broadcasting. 1 Introduction Unless stated otherwise each digraph of this paper is nite and may have loops but not multiarcs. Let G be a digraph. Then V (G) and A(G) denote the vertex set and the arc set of G, respectively. Let (u; v) 2 A(G). Then we say that u is adjacent to v, and v is adjacent from u. Also, it is said that (u; v) is incident to v and incident from u. Let (v; w) 2 A(G). Then we say that (u; v) is adjacent to (v; w), and (v; w) is adjacent from (u; v). Let ...
Independent Spanning Trees with Small Stretch Factors
, 1996
"... A pair of spanning trees rooted at a vertex r are independent if for every vertex v the pair of unique tree paths from v to the root r are disjoint. This paper presents the first analysis of the path lengths involved in independent spanning trees in 2edgeconnected and 2vertexconnected graphs. We ..."
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Cited by 1 (0 self)
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A pair of spanning trees rooted at a vertex r are independent if for every vertex v the pair of unique tree paths from v to the root r are disjoint. This paper presents the first analysis of the path lengths involved in independent spanning trees in 2edgeconnected and 2vertexconnected graphs. We present upper and lower bounds on the stretch factors of pairs of independent spanning trees, where the stretch factor of a spanning tree is defined to be the maximum ratio between the length of paths in the tree to the root to the length of the shortest path in the graph to the root. We prove that if G is a 2edgeconnected graph with the property that every edge lies on a cycle of size at most h than we can construct in linear time a pair of edgeindependent spanning trees whose stretch factors are bounded by O(h). In fact, we prove a more general result, namely that the stretch factor of both independent trees can be bounded by a minimax length of ears with respect to a certain class of ...
On the Problem of Scheduling Parallel Computations of Multibody Dynamic Analysis
 Transactions of ASME
, 1999
"... A formulation of a graph problem for scheduling parallel computations of multibody dynamic analysis is presented. The complexity of scheduling parallel computations for a multibody dynamic analysis is studied. The problem of finding a shortest critical branch spanning tree is described and transform ..."
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Cited by 1 (0 self)
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A formulation of a graph problem for scheduling parallel computations of multibody dynamic analysis is presented. The complexity of scheduling parallel computations for a multibody dynamic analysis is studied. The problem of finding a shortest critical branch spanning tree is described and transformed to a minimum radius spanning tree, which is solved by an algorithm of polynomial complexity. The problems of shortest critical branch minimum weight spanning tree (SCBMWST) and the minimum weight shortest critical branch spanning tree (MWSCBST) are also presented. Both problems are shown to be NPhard by proving that the bounded critical branch bounded weight spanning tree (BCBBWST) problem is NPcomplete. It is also shown that the minimum computational cost spanning tree (MCCST) is at least as hard as SCBMWST or MWSCBST problems, hence itself an NPhard problem. A heuristic approach to solving these problems is developed and implemented, and simulation results are discussed. 1 Introduct...
A Counterexample for the Proof of Implication Conjecture on Independent Spanning Trees
"... Abstract — Khuller and Schieber (1992) in [1] developed a constructive algorithm to prove that the existence of kvertex independent trees in a kvertex connected graph implies the existence of kedge independent trees in a kedge connected graph. In this paper, we show a counterexample where their ..."
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Abstract — Khuller and Schieber (1992) in [1] developed a constructive algorithm to prove that the existence of kvertex independent trees in a kvertex connected graph implies the existence of kedge independent trees in a kedge connected graph. In this paper, we show a counterexample where their algorithm fails. 1.
The Transformation of Optimal Independent Spanning Trees in Hypercubes
 THE 29TH WORKSHOP ON COMBINATORIAL MATHEMATICS AND COMPUTATION THEORY
"... Multiple spanning trees rooted at the same vertex, say r, of a given graph are said to be independent if for each nonroot vertex, say v, paths from r to v, one path in each spanning trees, are internally disjoint. It has been proved that there exist k optimal independent spanning trees (OIST for sh ..."
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Multiple spanning trees rooted at the same vertex, say r, of a given graph are said to be independent if for each nonroot vertex, say v, paths from r to v, one path in each spanning trees, are internally disjoint. It has been proved that there exist k optimal independent spanning trees (OIST for short) rooted at any vertex in the kdimensional hypercube. The word optimal is defined by an additional requirement: in each spanning tree, the distance from v to the only child of r must be the Hamming distance. In this paper, we shall propose an algorithm to generate (k − 1)! instances of OIST according to an existing OIST (including itself).