Results 1  10
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15
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.
Finding Four Independent Trees
"... Motivated by a multitree approach to the design of reliable communication protocols, Itai and Rodeh gave a linear time algorithm for finding two independent spanning trees in a 2connected graph. Cheriyan and Maheshwari gave an O(V  2) algorithm for finding three independent spanning trees in a 3 ..."
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Cited by 4 (0 self)
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Motivated by a multitree approach to the design of reliable communication protocols, Itai and Rodeh gave a linear time algorithm for finding two independent spanning trees in a 2connected graph. Cheriyan and Maheshwari gave an O(V  2) algorithm for finding three independent spanning trees in a 3connected graph. In this paper we present an O(V  3) algorithm for finding four independent spanning trees in a 4connected graph. We make use of chain decompositions of 4connected graphs. ∗ Partially supported by NSF VIGRE grant † Supported by CNPq (Proc: 200611/003) – Brazil ‡ Partially supported by NSF grant DMS0245530 and NSA grant MDA9040310052
Resilient multipath routing with independent directed acyclic graphs
 in Proc. IEEE TRANSACTION ON NETWORKING
, 2012
"... Abstract—In this paper, we introduce the concept of Independent Directed Acyclic Graphs (IDAGs) to achieve resilient multipath routing. Linkindependent (Nodeindependent) DAGs satisfy the property that any path from a source to the root on one DAG is linkdisjoint (nodedisjoint) with any path from ..."
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Cited by 4 (1 self)
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Abstract—In this paper, we introduce the concept of Independent Directed Acyclic Graphs (IDAGs) to achieve resilient multipath routing. Linkindependent (Nodeindependent) DAGs satisfy the property that any path from a source to the root on one DAG is linkdisjoint (nodedisjoint) with any path from the source to the root on the other DAG. Given a network, we develop polynomial time algorithms to compute linkindependent and nodeindependent DAGs. The algorithm developed in this paper: (1) achieves multipath routing; (2) guarantees recovery from single link failure; (3) utilizes all possible edges; and (4) achieves all these with at most one bit per packet as overhead when routing is based on destination address and incoming link. We demonstrate the effectiveness of the proposed IDAGs approach by comparing key performance indices to that of the independent trees technique. I.
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...
NonSeparating Cycles in 4Connected Graphs
, 2001
"... We prove that given any fixed edge ra in a 4connected graph G, there exists a cycle C through ra such that G − (V (C) − {r}) is 2connected. This will provide the first step in a decomposition for 4connected graphs. We also prove that for any given edge e in a 5connected graph G there exists an ..."
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Cited by 2 (2 self)
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We prove that given any fixed edge ra in a 4connected graph G, there exists a cycle C through ra such that G − (V (C) − {r}) is 2connected. This will provide the first step in a decomposition for 4connected graphs. We also prove that for any given edge e in a 5connected graph G there exists an induced cycle C through e in G such that G − V (C) is 2connected. This provides evidence for a conjecture of Lovász.
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 ...
Open Problems 15
"... F20.22> 6;6 and K 6;7 are 3choosable (and simplified the proof that K 5;8 is 3choosable), thereby completing the proof that n(3) = 14. They also proved n(k) k \Delta n(k \Gamma 2) + 2 k , which improves the previously known upper bounds (from m(k)) when k is even. Zolt'an Furedi proved that K ..."
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F20.22> 6;6 and K 6;7 are 3choosable (and simplified the proof that K 5;8 is 3choosable), thereby completing the proof that n(3) = 14. They also proved n(k) k \Delta n(k \Gamma 2) + 2 k , which improves the previously known upper bounds (from m(k)) when k is even. Zolt'an Furedi proved that K 5;13 is not 3choosable, so the remaining complete bipartite graphs for which 3choosability has not been settled are K 5;t for 9 t 12 and K 6;t for 8 t 10. [6] also observes that K k;t is kchoosable if and only if t ! k<F
Nonseparating planar chains in 4connected graphs
, 2002
"... In this paper, we describe an O(V (G)  2) algorithm for finding a “nonseparating planar chain” in a 4connected graph G, which will be used to decompose an arbitrary 4connected graph into “planar chains”. This work was motivated by the study of a multitree approach to reliability in distributed ..."
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In this paper, we describe an O(V (G)  2) algorithm for finding a “nonseparating planar chain” in a 4connected graph G, which will be used to decompose an arbitrary 4connected graph into “planar chains”. This work was motivated by the study of a multitree approach to reliability in distributed networks, as well as the study of nonseparating induced paths in highly connected graphs.
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.