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45
The Thickness of Graphs: A Survey
 Graphs Combin
, 1998
"... We give a stateoftheart survey of the thickness of a graph from both a theoretical and a practical point of view. After summarizing the relevant results concerning this topological invariant of a graph, we deal with practical computation of the thickness. We present some modifications of a ba ..."
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We give a stateoftheart survey of the thickness of a graph from both a theoretical and a practical point of view. After summarizing the relevant results concerning this topological invariant of a graph, we deal with practical computation of the thickness. We present some modifications of a basic heuristic and investigate their usefulness for evaluating the thickness and determining a decomposition of a graph in planar subgraphs. Key words: Thickness, maximum planar subgraph, branch and cut 1 Introduction In VLSI circuit design, a chip is represented as a hypergraph consisting of nodes corresponding to macrocells and of hyperedges corresponding to the nets connecting the cells. A chipdesigner has to place the macrocells on a printed circuit board (which usually consists of superimposed layers), according to several designing rules. One of these requirements is to avoid crossings, since crossings lead to undesirable signals. It is therefore desirable to find ways to handle wi...
On Achieving Optimized Capacity Utilization in Application Overlay Networks with Multiple Competing Sessions
 Sessions, 16th annual ACM symposium on parallelism in algorithms and architectures (SPAA ’04
, 2004
"... In this paper, we examine the problem of largevolume data dissemination via overlay networks. A natural way to maximize the throughput of an overlay multicast session is to split the traffic and feed them into multiple trees. While in singletree solutions, bandwidth of leaf nodes may remain larg ..."
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In this paper, we examine the problem of largevolume data dissemination via overlay networks. A natural way to maximize the throughput of an overlay multicast session is to split the traffic and feed them into multiple trees. While in singletree solutions, bandwidth of leaf nodes may remain largely underutilized, multitree solutions increase the chances for a node to contribute its bandwidth by being a relaying node in at least one of the trees. We study the following problems: (1) What is the maximum capacity multitree solutions can exploit from overlay networks? (2) When multiple sessions compete within the same network, what is the relationship of two contradictory goals: achieving fairness and maximizing overall throughput? (3) What is the impact of IP routing in achieving at constraining the optimal performance of overlay multicast? We extend the multicommodity flow model to the case of overlay data dissemination, where each commodity is associated with an overlay session, rather than the traditional sourcedestination pair. We first prove that the problem is solvable in polynomial time, then propose an #approximation algorithm, assuming that each commodity can be split in arbitrary ways. The solution to this problem establishes the theoretical upper bound of overall throughput that any multitree solution could reach. We then study the same problem with the restriction that each commodity can only be split and fed into a limited number of trees. A randomized rounding algorithm and an online treeconstruction algorithm are presented. All these algorithms are evaluated by extensive simulations.
Lower bounds on twoterminal network reliability
, 1985
"... One measure of twoterminal network reliability, termed probabilistic connectedness, is the probability that two specified communication centers can communicate. A standard model of a network is a graph in which nodes represent communications centers and edges represent links between communication c ..."
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Cited by 12 (0 self)
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One measure of twoterminal network reliability, termed probabilistic connectedness, is the probability that two specified communication centers can communicate. A standard model of a network is a graph in which nodes represent communications centers and edges represent links between communication centers. Edges are assumed to have statistically independent probabilities of failing and nodes are assumed to be perfectly reliable. Exact calculation of twoterminal reliability for general networks has been shown to be #Pcomplete. As a result is desirable to compute upper and lower bounds that avoid the exponential computation likely required by exact algorithms. Two methods are considered for computing lower bounds on twoterminal reliability
Characterizations of arboricity of graphs
 Ars Combinatorica
"... The aim of this paper is to give several characterizations for the following two classes of graphs: (i) graphs for which adding any l edges produces a graph which is decomposible into k spanning trees and (ii) graphs for which adding some l edges produces a graph which is decomposible into k spannin ..."
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The aim of this paper is to give several characterizations for the following two classes of graphs: (i) graphs for which adding any l edges produces a graph which is decomposible into k spanning trees and (ii) graphs for which adding some l edges produces a graph which is decomposible into k spanning trees. Introduction and Theorems The concept of decomposing a graph into the minimum number of trees or forests dates back to NashWilliams and Tutte [6, 7, 11]. Since then, many authors have examined various tree decompositions of classes of graphs (for example [2, 8]). The aim of this paper is to give several characterizations for
What is a matroid?
, 2007
"... Matroids were introduced by Whitney in 1935 to try to capture abstractly the essence of dependence. Whitney’s definition embraces a surprising diversity of combinatorial structures. Moreover, matroids arise naturally in combinatorial optimization since they are precisely the structures for which th ..."
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Cited by 10 (0 self)
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Matroids were introduced by Whitney in 1935 to try to capture abstractly the essence of dependence. Whitney’s definition embraces a surprising diversity of combinatorial structures. Moreover, matroids arise naturally in combinatorial optimization since they are precisely the structures for which the greedy algorithm works. This survey paper introduces matroid theory, presents some of the main theorems in the subject, and identifies some of the major problems of current research interest.
NonAdaptive Fault Diagnosis for AllOptical Networks via Combinatorial Group Testing on Graphs
"... Abstract—We consider the fault diagnosis problem in alloptical networks, focusing on probing schemes to detect faults. Our work concentrates on nonadaptive probing schemes, in order to meet the stringent time requirements for fault recovery. This fault diagnosis problem motivates a new technical fr ..."
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Cited by 8 (0 self)
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Abstract—We consider the fault diagnosis problem in alloptical networks, focusing on probing schemes to detect faults. Our work concentrates on nonadaptive probing schemes, in order to meet the stringent time requirements for fault recovery. This fault diagnosis problem motivates a new technical framework that we introduce: group testing with graphbased constraints. Using this framework, we develop several new probing schemes to detect network faults. The efficiency of our schemes often depends on the network topology; in many cases we can show that our schemes are nearoptimal by providing tight lower bounds. I.
Locally finite graphs with ends: a topological approach
"... This paper is intended as an introductory survey of a newly emerging field: a topological approach to the study of locally finite graphs that crucially incorporates their ends. Topological arcs and circles, which may pass through ends, assume the role played in finite graphs by paths and cycles. Thi ..."
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Cited by 6 (6 self)
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This paper is intended as an introductory survey of a newly emerging field: a topological approach to the study of locally finite graphs that crucially incorporates their ends. Topological arcs and circles, which may pass through ends, assume the role played in finite graphs by paths and cycles. This approach has made it possible to extend to locally finite graphs many classical theorems of finite graph theory that do not extend verbatim. The shift of paradigm it proposes is thus as much an answer to old questions as a source of new ones; many concrete problems of both types are suggested in the paper. This paper attempts to provide an entry point to this field for readers that have not followed the literature that has emerged in the last 10 years or so. It takes them on a quick route through what appear to be the most important lasting results, introduces them to key proof techniques, identifies the most promising open
Optimal Graph Orientation with Storage Applications
, 1995
"... We show that the edges of a graph with maximum edge density d can always be oriented such that each vertex has indegree at most d. Hence, for arbitrary graphs, edges can always be assigned to incident vertices as uniformly as possible. For example, indegree 3 is achieved for planar graphs. This im ..."
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Cited by 5 (1 self)
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We show that the edges of a graph with maximum edge density d can always be oriented such that each vertex has indegree at most d. Hence, for arbitrary graphs, edges can always be assigned to incident vertices as uniformly as possible. For example, indegree 3 is achieved for planar graphs. This immediately gives a spaceoptimal data structure that answers edge membership queries in a maximum edge densityd graph in O(log d) time. Keywords Graph orientation, edge density, Hall condition, balanced adjacency lists, edge membership queries 1 The Theorem Let G be an undirected graph with n vertices and m edges. The parameter ffi(G) = m n is commonly called the edge density of G. The maximum (edge) density is the smallest integer d such that the edge density of any subgraph of G does not exceed d. More precisely, d = dmaxfffi(G 0 ) j G 0 is a subgraph of Gge. For example, d 1 for trees, d 3 for planar graphs, d = d 1 2 log 2 ne for hypercubes [GG,AH], and d d 1 2 (c \Gamma...
A necessary condition for the existence of disjoint bases of a family of infinite matroids
 CONGR. NUMERANTIUM 105 (1994)
, 1994
"... Let M = (Mr)r∈R be a system of matroids on a set S. Following the ideas of NashWilliams [7], for every transfinite sequence f of distinct elements of S, we define a number η(f). We prove that the condition that η(f) ≥ 0 for every possible choice of f is necessary for M to have a system of mutually ..."
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Cited by 5 (2 self)
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Let M = (Mr)r∈R be a system of matroids on a set S. Following the ideas of NashWilliams [7], for every transfinite sequence f of distinct elements of S, we define a number η(f). We prove that the condition that η(f) ≥ 0 for every possible choice of f is necessary for M to have a system of mutually disjoint bases. Further, we show that this condition is sufficient if R is countable and Mr is a rankfinite transversal matroid for every r ∈ R. We also present conjectures about edgedisjoint spanning trees and detachments of countable graphs.
An Algorithm for Packing Connectors
, 1997
"... Given an undirected graph G = (V; E) and a partition fS; Tg of V , an ST connector is a set of edges F ` E such that every component of the subgraph (V; F ) intersects both S and T . If either S or T is a singleton, then an ST connector is a spanning tree of G. On the other hand, if G is bip ..."
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Cited by 3 (0 self)
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Given an undirected graph G = (V; E) and a partition fS; Tg of V , an ST connector is a set of edges F ` E such that every component of the subgraph (V; F ) intersects both S and T . If either S or T is a singleton, then an ST connector is a spanning tree of G. On the other hand, if G is bipartite with colour classes S and T , then an ST connector is an edge cover of G (a set of edges covering all vertices). An ST connector is a common spanning set of two graphic matroids on E. We prove a theorem on packing common spanning sets of certain matroids, generalizing a result of Davies and McDiarmid on strongly base orderable matroids. As a corollary, we obtain an O((n; m) + nm) time algorithm for finding a maximum number of ST connectors, where (n; m) denotes the complexity of finding a maximum number of edge disjoint spanning trees in a graph on n vertices and m edges. Since the best known bound for (n; m) is O(nm log(m=n)), this bound for packing ST connectors ...