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38
Algorithmic Graph Minor Theory: Decomposition, Approximation, and Coloring
 In 46th Annual IEEE Symposium on Foundations of Computer Science
, 2005
"... At the core of the seminal Graph Minor Theory of Robertson and Seymour is a powerful structural theorem capturing the structure of graphs excluding a fixed minor. This result is used throughout graph theory and graph algorithms, but is existential. We develop a polynomialtime algorithm using topolog ..."
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At the core of the seminal Graph Minor Theory of Robertson and Seymour is a powerful structural theorem capturing the structure of graphs excluding a fixed minor. This result is used throughout graph theory and graph algorithms, but is existential. We develop a polynomialtime algorithm using topological graph theory to decompose a graph into the structure guaranteed by the theorem: a cliquesum of pieces almostembeddable into boundedgenus surfaces. This result has many applications. In particular, we show applications to developing many approximation algorithms, including a 2approximation to graph coloring, constantfactor approximations to treewidth and the largest grid minor, combinatorial polylogarithmicapproximation to halfintegral multicommodity flow, subexponential fixedparameter algorithms, and PTASs for many minimization and maximization problems, on graphs excluding a fixed minor. 1.
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 37 (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.
Recent Excluded Minor Theorems for Graphs
 IN SURVEYS IN COMBINATORICS, 1999 267 201222. THE ELECTRONIC JOURNAL OF COMBINATORICS 8 (2001), #R34 8
, 1999
"... A graph is a minor of another if the first can be obtained from a subgraph of the second by contracting edges. An excluded minor theorem describes the structure of graphs with no minor isomorphic to a prescribed set of graphs. Splitter theorems are tools for proving excluded minor theorems. We disc ..."
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Cited by 9 (0 self)
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A graph is a minor of another if the first can be obtained from a subgraph of the second by contracting edges. An excluded minor theorem describes the structure of graphs with no minor isomorphic to a prescribed set of graphs. Splitter theorems are tools for proving excluded minor theorems. We discuss splitter theorems for internally 4connected graphs and for cyclically 5connected cubic graphs, the graph minor theorem of Robertson and Seymour, linkless embeddings of graphs in 3space, Hadwiger’s conjecture on tcolorability of graphs with no Kt+1 minor, Tutte’s edge 3coloring conjecture on edge 3colorability of 2connected cubic graphs with no Petersen minor, and Pfaffian orientations of bipartite graphs. The latter are related to the even directed circuit problem, a problem of Pólya about permanents, the 2colorability of hypergraphs, and signnonsingular matrices.
INDEPENDENT SETS IN GRAPHS WITH AN EXCLUDED CLIQUE MINOR
, 2006
"... Let G be a graph with n vertices, with independence number α, and with with no Kt+1minor for some t ≥ 5. It is proved that (2α − 1)(2t − 5) ≥ 2n − 5. ..."
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Cited by 7 (0 self)
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Let G be a graph with n vertices, with independence number α, and with with no Kt+1minor for some t ≥ 5. It is proved that (2α − 1)(2t − 5) ≥ 2n − 5.
Complete partitions of graphs
 In SODA
, 2005
"... A complete partition of a graph G is a partition of its vertex set in which any two distinct classes are connected by an edge. Let cp(G) denote the maximum number of classes in a complete partition of G. This measure was defined in 1969 by Gupta [G69], and is known to be NPhard to compute for sever ..."
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Cited by 5 (2 self)
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A complete partition of a graph G is a partition of its vertex set in which any two distinct classes are connected by an edge. Let cp(G) denote the maximum number of classes in a complete partition of G. This measure was defined in 1969 by Gupta [G69], and is known to be NPhard to compute for several classes of graphs. We obtain essentially tight lower and upper bounds on the approximability of this problem. We show that there is a randomized polynomialtime algorithm that given a graph G with n vertices, produces a complete partition of size Ω(cp(G) / √ lg n). This algorithm can be derandomized. We show that the upper bound is essentially tight: there is a constant C> 1, such that if there is a randomized polynomialtime algorithm that for all large n, when given a graph G with n vertices produces a complete partition into at least C · cp(G) / √ lg n classes, then NP ⊆ RTime(n O(lg lg n)). The problem of finding a complete partition of a graph is thus the first natural problem whose approximation threshold has been determined to be of the form Θ((lg n) c) for some constant c strictly between 0 and 1. 1
Minors in random and expanding hypergraphs
 in 27th SoCG’, ACM
, 2011
"... We introduce a new notion of minors for simplicial complexes (hypergraphs), socalled homological minors. Our motivation is to propose a general approach to attack certain extremal problems for sparse simplicial complexes and the corresponding threshold problems for random complexes. In this paper, ..."
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We introduce a new notion of minors for simplicial complexes (hypergraphs), socalled homological minors. Our motivation is to propose a general approach to attack certain extremal problems for sparse simplicial complexes and the corresponding threshold problems for random complexes. In this paper, we focus on threshold problems. The basic model for random complexes is the LinialMeshulam model X k (n, p). By definition, such a complex has n vertices, a complete (k − 1)dimensional skeleton, and every possible kdimensional simplex is chosen independently with probability p. We show that for every k, t ≥ 1, there is a constant C = C(k, t) such that for p ≥ C/n, the random complex X k (n, p) asymptotically almost surely contains K k t (the complete kdimensional complex on t vertices) as a homological minor. As corollary, the threshold for (topological) embeddability of X k (n, p) into R 2k is at p = Θ(1/n). The method can be extended to other models of random complexes (for which the lower skeleta are not necessarily complete) and also to more general Tverbergtype problems, where instead of continuous maps without doubly covered image points (embeddings), we consider maps without qfold covered image points.
Decomposition, approximation, and coloring of oddminorfree graphs
"... We prove two structural decomposition theorems about graphs excluding a fixed odd minor H, and show how these theorems can be used to obtain approximation algorithms for several algorithmic problems in such graphs. Our decomposition results provide new structural insights into oddHminorfree graph ..."
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Cited by 3 (1 self)
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We prove two structural decomposition theorems about graphs excluding a fixed odd minor H, and show how these theorems can be used to obtain approximation algorithms for several algorithmic problems in such graphs. Our decomposition results provide new structural insights into oddHminorfree graphs, on the one hand generalizing the central structural result from Graph Minor Theory, and on the other hand providing an algorithmic decomposition into two boundedtreewidth graphs, generalizing a similar result for minors. As one example of how these structural results conquer difficult problems, we obtain a polynomialtime 2approximation for vertex coloring in oddHminorfree graphs, improving on the previous O(V (H))approximation for such graphs and generalizing the previous 2approximation for Hminorfree graphs. The class of oddHminorfree graphs is a vast generalization of the wellstudied Hminorfree graph families and includes, for example, all bipartite graphs plus a bounded number of apices. OddHminorfree graphs are particularly interesting from a structural graph theory perspective because they break away from the sparsity of Hminorfree graphs, permitting a quadratic number of edges.
Hadwiger Number and the Cartesian Product of Graphs
 GRAPHS AND COMBINATORICS
, 2008
"... The Hadwiger number η(G) of a graph G is the largest integer n for which the complete graph Kn on n vertices is a minor of G. Hadwiger conjectured that for every graph G, η(G) ≥ χ(G), whereχ(G) is the chromatic number of G. Inthispaper,westudythe Hadwiger number of the Cartesian product G ✷ H of gr ..."
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Cited by 2 (0 self)
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The Hadwiger number η(G) of a graph G is the largest integer n for which the complete graph Kn on n vertices is a minor of G. Hadwiger conjectured that for every graph G, η(G) ≥ χ(G), whereχ(G) is the chromatic number of G. Inthispaper,westudythe Hadwiger number of the Cartesian product G ✷ H of graphs. As the main result of this paper, we prove that η(G 1 ✷ G 2) ≥ h √ l (1 − o(1)) for any two graphs G 1 and G 2 with η(G 1) = h and η(G 2) = l. We show that the above lower bound is asymptotically best possible when h ≥ l. This asymptotically settles a question of Z. Miller (1978). As consequences of our main result, we show the following: 1. Let G be a connected graph. Let G = G 1 ✷ G 2 ✷... ✷ Gk be the (unique) prime factorization of G.ThenG satisfies Hadwiger’s conjecture if k ≥ 2 log log χ(G) + c ′,wherec ′ is a constant. This improves the 2 log χ(G) + 3 bound in [2]. 2. Let G 1 and G 2 be two graphs such that χ(G 1) ≥ χ(G 2) ≥ c log 1.5 (χ(G 1)),wherec is a constant. Then G 1 ✷ G 2 satisfies Hadwiger’s conjecture. 3. Hadwiger’s conjecture is true for G d (Cartesian product of G taken d times) for every graph G and every d ≥ 2. This settles a question by Chandran and Sivadasan [2]. (They had shown that the Hadiwger’s conjecture is true for G d if d ≥ 3).
Hadwiger’s conjecture for proper circular arc graphs
"... Abstract. Circular arc graphs are graphs whose vertices can be represented as arcs on a circle such that any two vertices are adjacent if and only if their corresponding arcs intersect. Proper circular arc graphs are graphs which have a circular arc representation where no arc is completely containe ..."
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Abstract. Circular arc graphs are graphs whose vertices can be represented as arcs on a circle such that any two vertices are adjacent if and only if their corresponding arcs intersect. Proper circular arc graphs are graphs which have a circular arc representation where no arc is completely contained in any other arc. Hadwiger’s conjecture states that if a graph G has chromatic number k, then a complete graph of k vertices is a minor of G. We prove Hadwiger’s conjecture for proper circular arc graphs. Key words. circular arc, proper circular arc, Hadwiger’s conjecture, minor, graph coloring 1. Introduction. Circular