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33
A Polyhedral Approach to Planar Augmentation and Related Problems
, 1995
"... . Given a planar graph G, the planar (biconnectivity) augmentation problem is to add the minimum number of edges to G such that the resulting graph is still planar and biconnected. Given a nonplanar and biconnected graph, the maximum planar biconnected subgraph problem consists of removing the minim ..."
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Cited by 15 (1 self)
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. Given a planar graph G, the planar (biconnectivity) augmentation problem is to add the minimum number of edges to G such that the resulting graph is still planar and biconnected. Given a nonplanar and biconnected graph, the maximum planar biconnected subgraph problem consists of removing the minimum number of edges so that planarity is achieved and biconnectivity is maintained. Both problems are important in Automatic Graph Drawing. In [JM95], the minimum planarizing k augmentation problem has been introduced, that links the planarization step and the augmentation step together. Here, we are given a graph which is not necessarily planar and not necessarily kconnected, and we want to delete some set of edges D and to add some set of edges A such that jDj + jAj is minimized and the resulting graph is planar, kconnected and spanning. For all three problems, we have given a polyhedral formulation by defining three different linear objective functions over the same polytope, namely ...
Group Membership and Communication in Highly Mobile Ad Hoc Networks
, 2001
"... This thesis proposes the use of a new routing paradigm to enable communication in highly mobile, ad hoc networks, which operate wirelessly in the absence of dedicated master stations or fixed infrastructure. Due to the mobility of the nodes, the network topology changes frequently and unpredictably. ..."
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Cited by 14 (2 self)
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This thesis proposes the use of a new routing paradigm to enable communication in highly mobile, ad hoc networks, which operate wirelessly in the absence of dedicated master stations or fixed infrastructure. Due to the mobility of the nodes, the network topology changes frequently and unpredictably. We explore the new routing paradigm in the context of intervehicle communication. In such highly mobile ad hoc networks, the nodes commonly do not know the identity of their communication partners in advance. Rapid topology changes and scarce bandwidth prevent the nodes from exchanging updates regularly throughout the network. Therefore, we advocate a new routing paradigm that implicitly addresses message destinations based on the current situation of the network. [...]
EdgeRemoval and NonCrossing Configurations in Geometric Graphs
 In Proceedings of 24th European Conference on Computational Geometry
, 2008
"... A geometric graph is a graph G = (V, E) drawn in the plane, such that V is a point set in general position and E is a set of straightline segments whose endpoints belong to V. We study the following extremal problem for geometric graphs: How many arbitrary edges can be removed from a complete geome ..."
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Cited by 5 (2 self)
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A geometric graph is a graph G = (V, E) drawn in the plane, such that V is a point set in general position and E is a set of straightline segments whose endpoints belong to V. We study the following extremal problem for geometric graphs: How many arbitrary edges can be removed from a complete geometric graph with n vertices such that the remaining graph still contains a certain noncrossing subgraph. The noncrossing subgraphs that we consider are perfect matchings, subtrees of a given size, and triangulations. In each case, we obtain tight bounds on the maximum number of removable edges. 1
Undirected VertexConnectivity Structure and Smallest FourVertexConnectivity Augmentation
 Proc. 6th ISAAC
, 1995
"... In this paper, we study properties for the structure of an undirected graph that is not 4vertexconnected. We also study the evolution of this structure when an edge is added to optimally increase the vertexconnectivity of the underlying graph. Several properties reported here can be extended t ..."
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Cited by 4 (0 self)
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In this paper, we study properties for the structure of an undirected graph that is not 4vertexconnected. We also study the evolution of this structure when an edge is added to optimally increase the vertexconnectivity of the underlying graph. Several properties reported here can be extended to the case of a graph that is not kvertex connected, for an arbitrary k. Using properties obtained here, we solve the problem of finding a smallest set of edges whose addition 4vertexconnects an undirected graph. This is a fundamental problem in graph theory and has applications in network reliability and in statistical data security. We give an O(n \Delta log n + m)time algorithm for finding a set of edges with the smallest cardinality whose addition 4vertexconnects an undirected graph, where n and m are the number of vertices and edges in the input graph, respectively. This is the first polynomial time algorithm for this problem when the input graph is not 3vertexconnecte...
Editing Graphs into Disjoint Unions of Dense Clusters
"... Abstract. In the ΠCluster Editing problem, one is given an undirected graph G, a density measure Π, and an integer k ≥ 0, and needs to decide whether it is possible to transform G by editing (deleting and inserting) at most k edges into a dense cluster graph. Herein, a dense cluster graph is a grap ..."
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Cited by 4 (1 self)
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Abstract. In the ΠCluster Editing problem, one is given an undirected graph G, a density measure Π, and an integer k ≥ 0, and needs to decide whether it is possible to transform G by editing (deleting and inserting) at most k edges into a dense cluster graph. Herein, a dense cluster graph is a graph in which every connected component K = (VK, EK) satisfies Π. The wellstudied Cluster Editing problem is a special case of this problem with Π:=“being a clique”. In this work, we consider three other density measures that generalize cliques: 1) having at most s missing edges (sdefective cliques), 2) having average degree at least VK  − s (averagesplexes), and 3) having average degree at least µ · (VK  − 1) (µcliques), where s and µ are a fixed integer and a fixed rational number, respectively. We first show that the ΠCluster Editing problem is NPcomplete for all three density measures. Then, we study the fixedparameter tractability of the three clustering problems, showing that the first two problems are fixedparameter tractable with respect to the parameter (s, k) and that the third problem is W[1]hard with respect to the parameter k for 0 < µ < 1. 1
Hybrid Dissemination: Adding Determinism to Probabilistic Multicasting in LargeScale P2P Systems ⋆
"... Abstract. Epidemic protocols have demonstrated remarkable scalability and robustness in disseminating information on internetscale, dynamic P2P systems. However, popular instances of such protocols suffer from a number of significant drawbacks, such as increased message overhead in pushbased syste ..."
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Cited by 3 (1 self)
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Abstract. Epidemic protocols have demonstrated remarkable scalability and robustness in disseminating information on internetscale, dynamic P2P systems. However, popular instances of such protocols suffer from a number of significant drawbacks, such as increased message overhead in pushbased systems, or low dissemination speed in pullbased ones. In this paper we study pushbased epidemic dissemination algorithms, in terms of hit ratio, communication overhead, dissemination speed, and resilience to failures and node churn. We devise a hybrid pushbased dissemination algorithm, combining probabilistic with deterministic properties, which limits message overhead to an order of magnitude lower than that of the purely probabilistic dissemination model, while retaining strong probabilistic guarantees for complete dissemination of messages. Our extensive experimentation shows that our proposed algorithm outperforms that model both in static and dynamic network scenarios, as well as in the face of largescale catastrophic failures. Moreover, the proposed algorithm distributes the dissemination load uniformly on all participating nodes. Keywords: Epidemic/Gossip protocols, Information Dissemination, PeertoPeer. 1
COVER TIME AND BROADCAST TIME
, 2009
"... We introduce a new technique for bounding the cover time of random walks by relating it to the runtime of randomized broadcast. In particular, we strongly confirm for dense graphs the intuition of Chandra et al. [8] that “the cover time of the graph is an appropriate metric for the performance of c ..."
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We introduce a new technique for bounding the cover time of random walks by relating it to the runtime of randomized broadcast. In particular, we strongly confirm for dense graphs the intuition of Chandra et al. [8] that “the cover time of the graph is an appropriate metric for the performance of certain kinds of randomized broadcast algorithms”. In more detail, our results are as follows: • For any graph G = (V, E) of size n and minimum degree δ, we have R(G) = O ( E δ · log n), where R(G) denotes the quotient of the cover time and broadcast time. This bound is tight for binary trees and tight up to logarithmic factors for many graphs including hypercubes, expanders and lollipop graphs. • For any δregular (or almost δregular) graph G it holds that R(G) = Ω ( δ2 n · 1 log n). Together with our upper bound on R(G), this lower bound strongly confirms the intuition of Chandra et al. for graphs with minimum degree Θ(n), since then the cover time equals the broadcast time multiplied by n (neglecting logarithmic factors). • Conversely, for any δ we construct almost δregular graphs that satisfy R(G) = O(max { √ n, δ} · log 2 n). Since any regular expander satisfies R(G) = Θ(n), the strong relationship given above does not hold if δ is polynomially smaller than n. Our bounds also demonstrate that the relationship between cover time and broadcast time is much stronger than the known relationships between any of them and the mixing time (or the closely related spectral gap).
On the graphs with maximum distance on kdiameter
"... The distance of a set of vertices is the sum of the distances between pairs of vertices in the set. We define the kdiameter of a graph as the maximum distance of a set of k vertices; so the 2diameter is the normal diameter and the ndiameter where n is the order is the distance of the graph. We co ..."
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Cited by 2 (1 self)
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The distance of a set of vertices is the sum of the distances between pairs of vertices in the set. We define the kdiameter of a graph as the maximum distance of a set of k vertices; so the 2diameter is the normal diameter and the ndiameter where n is the order is the distance of the graph. We complete the characterisation of graphs with maximum distance given the order and size. We also determine the maximum size of a graph with given order and 3diameter. 1
Extremal Graphs In Connectivity Augmentation
, 1997
"... Let A(n; k; t) denote the smallest integer e for which every kconnected graph on n vertices can be made (k + t)connected by adding e new edges. We determine A(n; k; t) for all values of n; k and t in the case of (directed and undirected) edgeconnectivity and also for directed vertexconnectivity ..."
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Cited by 1 (1 self)
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Let A(n; k; t) denote the smallest integer e for which every kconnected graph on n vertices can be made (k + t)connected by adding e new edges. We determine A(n; k; t) for all values of n; k and t in the case of (directed and undirected) edgeconnectivity and also for directed vertexconnectivity. For undirected vertexconnectivity we determine A(n; k; 1) for all values of n and k. We also describe the graphs which attain the extremal values. 1 Introduction B. Bollob'as posed the following problem in his book Extremal Graph Theory in 1978, see [3, Page 49, Problem 34]. Let 1 k ! l ! n. Determine the minimal integer e for which to every kconnected graph of order n it is possible to add at most e edges such that the resulting graph is lconnected. Determine the analogous minimum for edgeconnectivity. In this paper we determine this number e for all values of n; k; l in the case of (directed and undirected) edgeconnectivity and directed vertexconnectivity. In the undirected ver...
Designing Reliable Architecture for Stateful Fault Tolerance
 In Proceedings of Seventh International Conference on Parallel and Distributed Computing, Applications and Technologies (PDCAT’06
, 2006
"... Performance and fault tolerance are two major issues that need to be addressed while designing highly available and reliable systems. The network topology or the notion of connectedness among the network nodes defines the system communication architecture and is an important design consideration for ..."
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Cited by 1 (1 self)
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Performance and fault tolerance are two major issues that need to be addressed while designing highly available and reliable systems. The network topology or the notion of connectedness among the network nodes defines the system communication architecture and is an important design consideration for fault tolerant systems. A number of fault tolerant designs for specific multiprocessor architecture exists in the literature, but none of them discriminates between stateless and stateful failover. In this paper, we propose a reliable network topology and a high availability framework which is tolerant upto a maximum of k node faults in a network and is designed specifically to meet the needs of stateful failover. Assuming the nodes in the network are capable of handling multiple processes, through our design we have been able to prove that in the event of k node failures the load can be uniformly distributed across the network ensuring load balance. We also provide an useful characterization for the network, which under the proposed framework ensures one hop communication between the required nodes.