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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.
Directional Routing via Generals stNumberings
 SIAM JOURNAL ON DISCRETE MATHEMATICS
, 2000
"... We present a mathematical model for network routing based on generating paths in a consistent direction. Our development is based on an algebraic and geometric framework for defining a directional coordinate system for real vector spaces. Our model, which generalizes graph stnumberings, is based on ..."
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Cited by 3 (1 self)
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We present a mathematical model for network routing based on generating paths in a consistent direction. Our development is based on an algebraic and geometric framework for defining a directional coordinate system for real vector spaces. Our model, which generalizes graph stnumberings, is based on mapping the nodes of a network to points in multidimensional space and ensures that the paths generated in di#erent directions from the same source are nodedisjoint. Such directional embeddings encode the global disjoint path structure with very simple local information. We prove that all 3connected graphs have 3directional embeddings in the plane so that each node outside a set of extreme nodes has a neighbor in each of the three directional regions defined in the plane. We conjecture that the result generalizes to kconnected graphs. We also showthat a directed acyclic graph (dag) that is kconnected to a set of sinks has a kdirectional embedding in (k  1)space with the sink set as the extreme nodes.
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 ...
Disjoint Paths in Arborescences
, 2004
"... An arborescence in a digraph is a tree directed away from its root. A classical theorem of Edmonds characterizes which digraphs have λ arcdisjoint arborescences rooted at r. A similar theorem of Menger guarantees λ strongly arc disjoint rvpaths for every vertex v, where “strongly ” means no two pa ..."
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An arborescence in a digraph is a tree directed away from its root. A classical theorem of Edmonds characterizes which digraphs have λ arcdisjoint arborescences rooted at r. A similar theorem of Menger guarantees λ strongly arc disjoint rvpaths for every vertex v, where “strongly ” means no two paths contain a pair of symmetric arcs. We prove that if a directed graph D contains two arcdisjoint spanning arborescences rooted at r, then D contains two such arborences with the property that for every node v the paths from r to v in the two arborences satisfy Menger‘s theorem.