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20
Hadwiger’s conjecture for K6free graphs
 COMBINATORICA
, 1993
"... In 1943, Hadwiger made the conjecture that every loopless graph not contractible to the complete graph on t + 1 vertices is tcolourable. When t ≤ 3 this is easy, and when t = 4, Wagner’s theorem of 1937 shows the conjecture to be equivalent to the fourcolour conjecture (the 4CC). However, when t ..."
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Cited by 34 (2 self)
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In 1943, Hadwiger made the conjecture that every loopless graph not contractible to the complete graph on t + 1 vertices is tcolourable. When t ≤ 3 this is easy, and when t = 4, Wagner’s theorem of 1937 shows the conjecture to be equivalent to the fourcolour conjecture (the 4CC). However, when t ≥ 5 it has remained open. Here we show that when t = 5 it is also equivalent to the 4CC. More precisely, we show (without assuming the 4CC) that every minimal counterexample to Hadwiger’s conjecture when t = 5 is “apex”, that is, it consists of a planar graph with one additional vertex. Consequently, the 4CC implies Hadwiger’s conjecture when t = 5, because it implies that apex graphs are 5colourable.
Detecting a Network Failure
 Proc. 41st Annual IEEE Symposium on Foundations of Computer Science
, 2000
"... Abstract. Measuring the properties of a large, unstructured network can be difficult: One may not have full knowledge of the network topology, and detailed global measurements may be infeasible. A valuable approach to such problems is to take measurements from selected locations within the network a ..."
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Cited by 33 (1 self)
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Abstract. Measuring the properties of a large, unstructured network can be difficult: One may not have full knowledge of the network topology, and detailed global measurements may be infeasible. A valuable approach to such problems is to take measurements from selected locations within the network and then aggregate them to infer largescale properties. One sees this notion applied in settings that range from Internet topology discovery tools to remote software agents that estimate the download times of popular web pages. Some of the most basic questions about this type of approach, however, are largely unresolved at an analytical level. How reliable are the results? How much does the choice of measurement locations affect the aggregate information one infers about the network? We describe algorithms that yield provable guarantees for a particular problem of this type: detecting a network failure. Suppose we want to detect events of the following form in an nnode network: An adversary destroys up to k nodes or edges, after which two subsets of the nodes, each of size at least εn, are disconnected from one another. We call such an event an (ε,k)partition. One method for detecting such events would be to place “agents ” at a set D of nodes, and record a fault whenever two of them become separated from each other. To be a good detection set, D should become disconnected whenever there is an (ε,k)partition; in this way, it “witnesses ” all such events. We show that every graph has a detection set of size polynomial in k and ε −1,and independent of the size of the graph itself. Moreover, random sampling provides an effective way to construct such a set. Our analysis establishes a connection between graph separators and the notion of VCdimension, using techniques based on matchings and disjoint paths.
A short proof of Mader's Spaths theorem
"... For an undirected graph G = (V; E) and a collection S of disjoint subsets of V , an Spath is a path connecting different sets in S. We give a short proof of Mader's minmax theorem for the maximum number of disjoint Spaths. Let G = (V; E) be an undirected graph and let S be a collection of disjoi ..."
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Cited by 12 (1 self)
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For an undirected graph G = (V; E) and a collection S of disjoint subsets of V , an Spath is a path connecting different sets in S. We give a short proof of Mader's minmax theorem for the maximum number of disjoint Spaths. Let G = (V; E) be an undirected graph and let S be a collection of disjoint subsets of V . An Spath is a path connecting two different sets in S. Mader [4] gave the following minmax relation for the maximum number of (vertex)disjoint Spaths, where S := S S.
The PathPacking Structure of Graphs
, 2003
"... We prove EdmondsGallai type structure theorems for Mader's edge and vertexdisjoint paths including also capacitated variants, and state a conjecture generalizing Mader's minimax theorems on path packings and Cunningham and Geelen's pathmatching theorem. ..."
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Cited by 3 (0 self)
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We prove EdmondsGallai type structure theorems for Mader's edge and vertexdisjoint paths including also capacitated variants, and state a conjecture generalizing Mader's minimax theorems on path packings and Cunningham and Geelen's pathmatching theorem.
Algebraic Algorithms for Linear Matroid Parity Problems
"... We present fast and simple algebraic algorithms for the linear matroid parity problem and its applications. For the linear matroid parity problem, we obtain a simple randomized algorithm with running time O(mrω−1) where m and r are the number of columns and the number of rows and ω ≈ 2.376 is the ma ..."
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Cited by 3 (1 self)
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We present fast and simple algebraic algorithms for the linear matroid parity problem and its applications. For the linear matroid parity problem, we obtain a simple randomized algorithm with running time O(mrω−1) where m and r are the number of columns and the number of rows and ω ≈ 2.376 is the matrix multiplication exponent. This improves the O(mrω)time algorithm by Gabow and Stallmann, and matches the running time of the algebraic algorithm for linear matroid intersection, answering a question of Harvey. We also present a very simple alternative algorithm with running time O(mr2) which does not need fast matrix multiplication. We further improve the algebraic algorithms for some specific graph problems of interest. For the Mader’s disjoint Spath problem, we present an O(nω)time randomized algorithm where n is the number of vertices. This improves the running time of the existing results considerably, and matches the running time of the algebraic algorithms for graph matching. For the graphic matroid parity problem, we give an O(n4)time randomized algorithm where n is the number of vertices, and an O(n3)time randomized algorithm for a special case useful in designing approximation algorithms. These algorithms are optimal in terms of n as the input size could be Ω(n4) and Ω(n3) respectively. The techniques are based on the algebraic algorithmic framework developed by Mucha and Sankowski, Harvey, and Sankowski. While linear matroid parity and Mader’s disjoint Spath are challenging generalizations for the design of combinatorial algorithms, our results show that both the algebraic algorithms for linear matroid intersection and graph matching can be extended nicely to more general settings. All algorithms are still faster than the existing algorithms even if fast matrix multiplication is not used. These provide simple algorithms that can be easily implemented in practice.
Toughness and Edgetoughness
 Discrete Mathematics
, 1997
"... this paper only finite, undirected and simple graphs are considered. In 1973 Chv'atal [4] introduced the definition of toughness. Many authors investigated the relation of toughness and hamiltonicity since Chv'atal's paper. A good survey of the topic is [2]. Let !(G) denote the number of components ..."
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Cited by 2 (2 self)
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this paper only finite, undirected and simple graphs are considered. In 1973 Chv'atal [4] introduced the definition of toughness. Many authors investigated the relation of toughness and hamiltonicity since Chv'atal's paper. A good survey of the topic is [2]. Let !(G) denote the number of components of a graph G. If T ` V (G) then the graph G \Gamma T is defined as follows. V (G \Gamma T ) = V (G) \Gamma T and (u; v) = e 2 E(G \Gamma T ) iff e 2 E(G) but non of u and v is in T . A graph G is ttough if jSj t!(G \Gamma S) for every subset S of the vertex set V (G) with
On a MinMax Theorem of Cacti
, 1997
"... A simple proof is presented for the minmax theorem of Lov'asz on cacti. Instead of using the result of Lov'asz on matroid parity, we shall apply twice the (conceptionally simpler) matroid intersection theorem. 1 Introduction The graph matching problem and the matroid intersection problem are two w ..."
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A simple proof is presented for the minmax theorem of Lov'asz on cacti. Instead of using the result of Lov'asz on matroid parity, we shall apply twice the (conceptionally simpler) matroid intersection theorem. 1 Introduction The graph matching problem and the matroid intersection problem are two wellsolved problems in Combinatorial Theory in the sense of minmax theorems and polynomial algorithms for finding an optimal solution. The matroid parity problem, a common generalization of them, turned out to be much more difficult. For the general problem there does not exist polynomial algorithm [2], [3]. Moreover, it contains NPcomplete problems. On the other hand, for linear matroids Lov'asz [3] provided a minmax formula and a polynomial algorithm. There are several earlier results which can be derived from Lov'asz' theorem, e.g. Tutte's result on ffactors [8], a result of Mader on openly disjoint Apaths [5], a result of Nebesky concerning maximum genus of graphs [6]. Another appli...
Packing nonreturning Apaths algorithmically
, 2005
"... In this paper we present an algorithmic approach to packing Apaths. It is regarded as a generalization of Edmonds’ matching algorithm, however there is the significant difference that here we do not build up any kind of alternating tree. Instead we use the socalled 3way lemma, which either provid ..."
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In this paper we present an algorithmic approach to packing Apaths. It is regarded as a generalization of Edmonds’ matching algorithm, however there is the significant difference that here we do not build up any kind of alternating tree. Instead we use the socalled 3way lemma, which either provides augmentation, or a dual, or a subgraph which can be used for contraction. The method works in the general setting of packing nonreturning Apaths. It also implies an eardecomposition of criticals, as a generalization of the odd eardecomposition of factorcritical graph.
Halfintegrality of nodecapacitated multiflows and treeshaped facility locations on trees
, 2010
"... In this paper, we establish a novel duality relationship between nodecapacitated multiflows and treeshaped facility locations. We prove that the maximum value of a treedistanceweighted maximum nodecapacitated multiflow problem is equal to the minimum value of the problem of locating subtrees in ..."
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Cited by 2 (2 self)
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In this paper, we establish a novel duality relationship between nodecapacitated multiflows and treeshaped facility locations. We prove that the maximum value of a treedistanceweighted maximum nodecapacitated multiflow problem is equal to the minimum value of the problem of locating subtrees in a tree, and the maximum is attained by a halfintegral multiflow. Utilizing this duality, we show that a halfintegral optimal multiflow and an optimal location can be found in strongly polynomial time. These extend previously known results in the maximum free multiflow problems. We also show that the set of treedistance weights is the only class having bounded fractionality in maximum nodecapacitated multiflow problems.
Lengthconstrained Pathmatchings in Graphs
, 2002
"... The pathmatching problem is to find a set of vertex or edge disjoint paths with length constraints in a given graph with a given set of endpoints. This problem has application in broadcasting and multicasting in computer networks. In this paper, we study the algorithmic complexity of different case ..."
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Cited by 1 (1 self)
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The pathmatching problem is to find a set of vertex or edge disjoint paths with length constraints in a given graph with a given set of endpoints. This problem has application in broadcasting and multicasting in computer networks. In this paper, we study the algorithmic complexity of different cases of this problem. In each case, we either provide a polynomial time algorithm or prove that the problem is NPcomplete.