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The MultiTree Approach to Reliability in Distributed Networks
 Information and Computation
, 1984
"... Consider a network of asynchronous processors communicating by sending messages over unreliable lines. There are many advantages to restricting all communications to a spanning tree. To overcome the possible failure of k
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Consider a network of asynchronous processors communicating by sending messages over unreliable lines. There are many advantages to restricting all communications to a spanning tree. To overcome the possible failure of k <k edges, we describe a communication protocol which uses k rooted spanning trees having the property that for every vertex v the paths from v to the root are edgedisjoint. An algorithm to find two such trees in a 2 edgeconnected graph is described that runs in time proportional to the number of edges in the graph. This algorithm has a distributed version which finds the two trees even when a single edge fails during their construction. The two trees them may be used to transform certain centralized algorithms to distributed, reliable and efficient ones.  1  1. INTRODUCTION Consider a network G=(V ,E ) of n = V asynchronous processors (or vertices) connected by e = E edges. The network may be used to conduct a computation which cannot be done in a single pr...
Deterministic O(nm) Time EdgeSplitting in Undirected Graphs
 J. Combinatorial Optimization
, 1997
"... This paper presents a deterministic O(nm log n + n 2 log 2 n) = ~ O(nm) time algorithm for splitting o all edges incident to a vertex s of even degree in a multigraph G, where n and m are the numbers of vertices and links (= vertex pairs between which G has an edge) in G, respectively. Based o ..."
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This paper presents a deterministic O(nm log n + n 2 log 2 n) = ~ O(nm) time algorithm for splitting o all edges incident to a vertex s of even degree in a multigraph G, where n and m are the numbers of vertices and links (= vertex pairs between which G has an edge) in G, respectively. Based on this, many graph algorithms using edgesplitting can run faster. For example, the edgeconnectivity augmentation problem in an undirected multigraph can be solved in ~ O(nm) time, which is an improvement over the previously known randomized ~ O(n 3 ) bound and deterministic ~ O(n 2 m) bound. 1 Introduction Let G = (V; E) stand for an undirected multigraph with a set V of vertices and a set E of edges, where an edge with end vertices u and v is denoted by (u; v). A singleton set fxg may be simply written as x, and \ " implies proper inclusion while \ " means \ " or \ = ". For two disjoint subsets X;Y V , we denote by EG (X; Y ) the set of edges, one of whose end vertices is i...
Twoconnected orientations of Eulerian graphs
 J. Graph Theory. Egres Technical Reports
, 2004
"... A graph G=(V,E) is said to be weakly fourconnected if G is 4edgeconnected and Gx is 2edgeconnected for every x \in V. We prove that every weakly fourconnected Eulerian graph has a 2connected Eulerian orientation. This verifies a special case of a conjecture of A. Frank. ..."
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A graph G=(V,E) is said to be weakly fourconnected if G is 4edgeconnected and Gx is 2edgeconnected for every x \in V. We prove that every weakly fourconnected Eulerian graph has a 2connected Eulerian orientation. This verifies a special case of a conjecture of A. Frank.
A Characterisation of Weakly FourConnected Graphs
, 2003
"... A graph G=(V,E) is called weakly fourconnected if G is 4edgeconnected and Gx is 2edgeconnected for all x \in V . We give sufficient conditions for the existence of `splittable' vertices of degree four in weakly fourconnected graphs. By using these results we prove that every minimally weakly ..."
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A graph G=(V,E) is called weakly fourconnected if G is 4edgeconnected and Gx is 2edgeconnected for all x \in V . We give sufficient conditions for the existence of `splittable' vertices of degree four in weakly fourconnected graphs. By using these results we prove that every minimally weakly fourconnected graph on at least four vertices contains at least three `splittable' vertices of degree four, which gives rise to an inductive construction of weakly fourconnected graphs. Our results can also be applied in the problem of finding 2connected orientations of graphs.
A Note on Mixed Graphs and Directed Splitting Off
, 1994
"... We give counterexamples to two conjectures in "Bill Jackson, Some remarks on arcconnectivity, vertex splitting, and orientation in graphs and digraphs, Journal of Graph Theory, 12(3):429436, 1988" concerning orientations of mixed graphs and splitting off in digraphs, and prove the first conjectur ..."
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We give counterexamples to two conjectures in "Bill Jackson, Some remarks on arcconnectivity, vertex splitting, and orientation in graphs and digraphs, Journal of Graph Theory, 12(3):429436, 1988" concerning orientations of mixed graphs and splitting off in digraphs, and prove the first conjecture in the (di) Eulerian case(s). Beside that we solve the degree constrained directed augmentation problem for diEulerian mixed graphs. 1 Introduction A mixed graph G = (V; A [ E) has vertex set V , undirected edge set E, and arc set A. We allow multiple edges but not loops. We write an arc directed from u to v as u!v and an edge with end nodes u and v as uv. For a set X ae V we let d(X) (d G (X) if we want to emphasize the graph we study) denote the number of edges between X and V \Gamma X, for a vertex v we call d(fvg) = d(v) the degree of v. %(X) is the number of arcs directed from V \Gamma X to X, ffi(X) = %(V \Gamma X). An orientation of G is an orientation of the edge set E and we l...
MATHEMATICAL ENGINEERING TECHNICAL REPORTS Recent Results on WellBalanced Orientations
, 2006
"... scholarly and technical work on a noncommercial basis. Copyright and all rights therein are maintained by the authors or by other copyright holders, notwithstanding that they have offered their works here electronically. It is understood that all persons copying this information will adhere to the ..."
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scholarly and technical work on a noncommercial basis. Copyright and all rights therein are maintained by the authors or by other copyright holders, notwithstanding that they have offered their works here electronically. It is understood that all persons copying this information will adhere to the terms and constraints invoked by each author’s copyright. These works may not be reposted without the explicit permission of the copyright holder. Recent results on wellbalanced orientations