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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. ..."
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Cited by 16 (3 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 6 (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.
Packing Arborescences
 RIMS KOKYUROKU BESSATSU(2010), B23: 131
, 2010
"... In [7], Edmonds proved a fundamental theorem on packing arborescences that has become the base of several subsequent extensions. Recently, Japanese researchers found an unexpected further generalization which gave rise to many interesting questions about this subject [29], [20]. Another line of rese ..."
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Cited by 2 (0 self)
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In [7], Edmonds proved a fundamental theorem on packing arborescences that has become the base of several subsequent extensions. Recently, Japanese researchers found an unexpected further generalization which gave rise to many interesting questions about this subject [29], [20]. Another line of researches focused on covering intersecting families which generalizes Edmonds ' theorems in a dierent way. The two approaches was united in [1] by introducing the notion of mixed intersecting biset families. The purpose of this paper is to overview recent developments and to present some new results. We give a polyhedral description of arborescence packable subgraphs based on a connection with biset families, and by using this description we prove TDIness of the corresponding system of inequalities. We also consider the problem of independent trees and arborescences, and give a simple algorithm that decomposes a maximal planar graph into three independent trees.
Parallel Construction of Independent Spanning Trees on Multidimensional Tori
 THE 24TH WORKSHOP ON COMBINATORIAL MATHEMATICS AND COMPUTATION THEORY
"... A set of spanning trees rooted at vertex r in G is called independent spanning trees (IST) if for each vertex v in G, v = r, the paths from v to r in any two trees are different and vertexdisjoint. If the connectivity of G is k, the IST problem is to construct k IST rooted at each vertex. The IST ..."
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A set of spanning trees rooted at vertex r in G is called independent spanning trees (IST) if for each vertex v in G, v = r, the paths from v to r in any two trees are different and vertexdisjoint. If the connectivity of G is k, the IST problem is to construct k IST rooted at each vertex. The IST problem has found applications in faulttolerant broadcasting, but it is still open for general graph with connectivity greater than four. Obokata et al. [IEICE Trans. Fundamentals of Electronics, Communications and Computer Sciences E79A (1996) 1894–1903] have proved that the IST problem can be solved on multidimensional tori. However, their construction algorithm forbids the possibility of parallel processing. In this paper, we shall propose a parallel algorithm that is based on the Latin square scheme to solve the IST problem on multidimensional tori.
The Mondshein Sequence
"... Canonical orderings [STOC’88, FOCS’92] have been used as a key tool in graph drawing, graph encoding and visibility representations for the last decades. We study a farreaching generalization of canonical orderings to nonplanar graphs that was published by Lee Mondshein in a PhDthesis at M.I.T. a ..."
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Canonical orderings [STOC’88, FOCS’92] have been used as a key tool in graph drawing, graph encoding and visibility representations for the last decades. We study a farreaching generalization of canonical orderings to nonplanar graphs that was published by Lee Mondshein in a PhDthesis at M.I.T. as early as 1971. Mondshein proposed to order the vertices of a graph in a sequence such that, for any i, the vertices from 1 to i induce essentially a 2connected graph while the remaining vertices from i + 1 to n induce a connected graph. Mondshein’s sequence generalizes canonical orderings and became later and independently known under the name nonseparating ear decomposition. Currently, the best known algorithm for computing this sequence achieves a running time of O(nm); the main open problem in Mondshein’s and followup work is to improve this running time to a subquadratic time. In this paper, we present the first algorithm that computes a Mondshein sequence in time and space O(m), improving the previous best running time by a factor of n. In addition, we illustrate the impact of this result by deducing lineartime algorithms for several other problems, for which the previous best running times have been quadratic. In particular, we show how to compute three independent spanning trees in a 3connected graph in linear time, improving a result of Cheriyan and Maheshwari [J. Algorithms 9(4)]. Secondly, we improve the preprocessing time for the outputsensitive data structure by Di Battista, Tamassia and Vismara [Algorithmica 23(4)] that reports three internally disjoint paths between any given vertex pair from O(n2) to O(m). Finally, we show how a very simple lineartime planarity test can be derived once a Mondshein sequence is computed.
Constructive Algorithm of Independent Spanning Trees on Möbius Cubes
, 2011
"... Independent spanning trees (ISTs) on networks have applications in networks such as reliable communication protocols, the multinode broadcasting, onetoall broadcasting, reliable broadcasting and secure message distribution. However, there is a problem on ISTs on graphs: If a graph G is nconnecte ..."
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Independent spanning trees (ISTs) on networks have applications in networks such as reliable communication protocols, the multinode broadcasting, onetoall broadcasting, reliable broadcasting and secure message distribution. However, there is a problem on ISTs on graphs: If a graph G is nconnected (n ≥ 1), then there are n ISTs rooted at an arbitrary vertex on G. This problem has remained open for n ≥ 5. In this paper, we consider the construction of ISTs on Möbius cubes—a class of hypercube variants. An O(N log N) recursive algorithm is proposed to construct n ISTs rooted at an arbitrary vertex on the ndimensional Möbius cube Mn, where N = 2n is the number of vertices in Mn. Furthermore, we prove that each IST obtained by our algorithm is isomorphic to an nlevel binomiallike tree with the height n + 1 for n ≥ 2.