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47
Haplotyping as Perfect Phylogeny: Conceptual Framework and Efficient Solutions (Extended Abstract)
, 2002
"... The next highpriority phase of human genomics will involve the development of a full Haplotype Map of the human genome [12]. It will be used in largescale screens of populations to associate specific haplotypes with specific complex geneticinfluenced diseases. A prototype Haplotype Mapping strat ..."
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Cited by 109 (10 self)
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The next highpriority phase of human genomics will involve the development of a full Haplotype Map of the human genome [12]. It will be used in largescale screens of populations to associate specific haplotypes with specific complex geneticinfluenced diseases. A prototype Haplotype Mapping strategy is presently being finalized by an NIH workinggroup. The biological key to that strategy is the surprising fact that genomic DNA can be partitioned into long blocks where genetic recombination has been rare, leading to strikingly fewer distinct haplotypes in the population than previously expected [12, 6, 21, 7]. In this paper
Homogeneous multivariate polynomials with the halfplane property
 Adv. in Appl. Math
"... A polynomial P in n complex variables is said to have the “halfplane property” (or Hurwitz property) if it is nonvanishing whenever all the variables lie in the open right halfplane. Such polynomials arise in combinatorics, reliability theory, electrical circuit theory and statistical mechanics. A ..."
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Cited by 39 (4 self)
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A polynomial P in n complex variables is said to have the “halfplane property” (or Hurwitz property) if it is nonvanishing whenever all the variables lie in the open right halfplane. Such polynomials arise in combinatorics, reliability theory, electrical circuit theory and statistical mechanics. A particularly important case is when the polynomial is homogeneous and multiaffine: then it is the (weighted) generating polynomial of an runiform set system. We prove that the support (set of nonzero coefficients) of a homogeneous multiaffine polynomial with the halfplane property is necessarily the set of bases of a matroid. Conversely, we ask: For which matroids M does the basis generating polynomial P B(M) have the halfplane property? Not all matroids have the halfplane property, but we find large classes that do: all sixthrootofunity matroids, and a subclass of transversal (or cotransversal) matroids that we call “nice”. Furthermore, the class of matroids with the halfplane property is closed under minors, duality, direct sums, 2sums, series and parallel connection, fullrank matroid union, and some special cases of principal truncation, principal extension, principal cotruncation and principal coextension. Our positive results depend on two distinct (and apparently unrelated) methods for constructing polynomials with the halfplane property: a determinant construction (exploiting “energy” arguments), and a permanent construction (exploiting the Heilmann–Lieb theorem on matching polynomials). We conclude with a list of open questions. KEY WORDS: Graph, matroid, jump system, abstract simplicial complex, spanning tree, basis, generating polynomial, reliability polynomial, Brown–Colbourn conjecture,
Deltamatroids, Jump Systems and Bisubmodular Polyhedra
, 1993
"... We relate an axiomatic generalization of matroids, called a jump system, to polyhedra arising from bisubmodular functions. Unlike the case for usual submodularity, the points of interest are not all the integral points in the relevant polyhedron, but form a subset of them. However, we do show that t ..."
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Cited by 33 (0 self)
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We relate an axiomatic generalization of matroids, called a jump system, to polyhedra arising from bisubmodular functions. Unlike the case for usual submodularity, the points of interest are not all the integral points in the relevant polyhedron, but form a subset of them. However, we do show that the convex hull of the set of points of a jump system is a bisubmodular polyhedron, and that the integral points of an integral bisubmodular polyhedron determine a (special) jump system. We also prove addition and composition theorems for jump systems, which have several applications for deltamatroids and matroids. Copyright (C) by the Society for Industrial and Applied Mathematics, in SIAM Journal on Discrete Mathematics, 8 (1995) pp. 1732. y Partially supported by an N.S.E.R.C. International Scientific Exchange Award at Carleton University z Partially supported by an N.S.E.R.C. of Canada operating grant 1 Introduction Matroids are important as a unifying structure in pure combin...
Decomposition of Balanced Matrices
 J. COMBINATORIAL THEORY, SER. B
, 1999
"... A 0,1 matrix is balanced if it does not contain a square submatrix of odd order with two ones per row and per column. We show that a balanced 0,1 matrix is either totally unimodular or its bipartite representation has a cutset consisting of two adjacent nodes and some of their neighbors. This resul ..."
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Cited by 29 (5 self)
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A 0,1 matrix is balanced if it does not contain a square submatrix of odd order with two ones per row and per column. We show that a balanced 0,1 matrix is either totally unimodular or its bipartite representation has a cutset consisting of two adjacent nodes and some of their neighbors. This result yields a polytime recognition algorithm for balancedness. To prove the result, we first prove a decomposition theorem for balanced 0,1 matrices that are not strongly balanced.
List Partitions
 Proc. 31st Ann. ACM Symp. on Theory of Computing
, 2003
"... List partitions generalize list colourings and list homomorphisms. Each symmetric matrix M over 0; 1; defines a list partition problem. Different choices of the matrix M lead to many wellknown graph theoretic problems including the problem of recognizing split graphs and their generalizations, ..."
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Cited by 27 (11 self)
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List partitions generalize list colourings and list homomorphisms. Each symmetric matrix M over 0; 1; defines a list partition problem. Different choices of the matrix M lead to many wellknown graph theoretic problems including the problem of recognizing split graphs and their generalizations, finding homogeneous sets, joins, clique cutsets, stable cutsets, skew cutsets and so on. We develop tools which allow us to classify the complexity of many list partition problems and, in particular, yield the complete classification for small matrices M . Along the way, we obtain a variety of specific results including: generalizations of Lov'asz's communication bound on the number of cliqueversus stableset separators; polynomialtime algorithms to recognize generalized split graphs; a polynomial algorithm for the list version of the Clique Cutset Problem; and the first subexponential algorithm for the Skew Cutset Problem of Chv'atal. We also show that the dichotomy (NP complete versus polynomialtime solvable), conjectured for certain graph homomorphism problems would, if true, imply a slightly weaker dichotomy (NP complete versus quasipolynomial) for our list partition problems 1 . Email: tomas@theory.stanford.edu. y School of Computing Science, Simon Fraser University, Burnaby, B.C., Canada, V5A1S6. Email: pavol@cs.sfu.ca. Supported by a Research Grant from the National Sciences and Engineering Research Council. z Departamento da Ciencia da Computac~ao  I.M., COPPE/Sistemas, Universidade Federal do Rio de Janeiro, RJ, 21945970, Brasil. Email: sula@cos.ufrj.br. Supported by CNPq and PRONEX 107/97. x Department of Computer Science, Stanford University, CA 943059045. Email: rajeev@cs.stanford.edu. Supported by an ARO MURI Grant DAAH04961...
The structure of the 3separations of 3connected matroids
 II, European J. Combin
"... Abstract. Tutte defined a k–separation of a matroid M to be a partition (A, B) of the ground set of M such that A, B  ≥ k and r(A) +r(B) − r(M)
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Cited by 18 (13 self)
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Abstract. Tutte defined a k–separation of a matroid M to be a partition (A, B) of the ground set of M such that A, B  ≥ k and r(A) +r(B) − r(M) <k. If, for all m<n, the matroid M has no m–separations, then M is n–connected. Earlier, Whitney showed that (A, B) is a 1–separation of M if and only if A is a union of 2–connected components of M. WhenMis2–connected, Cunningham and Edmonds gave a tree decomposition of M that displays all of its 2–separations. When M is 3–connected, this paper describes a tree decomposition of M that displays, up to a certain natural equivalence, all nontrivial 3– separations of M. 1.
A Matroid Invariant via the KTheory of the Grassmannian
, 2006
"... Let G(d,n) denote the Grassmannian of dplanes in Cn and let T be the torus (C∗) n /diag(C ∗ ) which acts on G(d,n). Let x be a point of G(d,n) and let Tx be the closure of the Torbit through x. Then the class of the structure sheaf of Tx in the Ktheory of G(d,n) depends only on which Plücker coor ..."
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Cited by 12 (1 self)
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Let G(d,n) denote the Grassmannian of dplanes in Cn and let T be the torus (C∗) n /diag(C ∗ ) which acts on G(d,n). Let x be a point of G(d,n) and let Tx be the closure of the Torbit through x. Then the class of the structure sheaf of Tx in the Ktheory of G(d,n) depends only on which Plücker coordinates of x are nonzero – combinatorial data known as the matroid of x. In this paper, we will define a certain map of additive groups from K ◦ (G(d,n)) to Z[t]. Letting gx(t) denote the image of (−1) n−dim Tx [OTx], gx behaves nicely under the standard constructions of matroid theory. Specifically, gx1⊕x2 (t) = gx1 (t)gx2(t), gx1+2 x2 (t) = gx1 (t)gx2(t)/t, gx(t) = gx⊥(t) and gx is unaltered by series and parallel extensions. Furthermore, the coefficients of gx are nonnegative. The existence of this map implies bounds on (essentially equivalently) the complexity of Kapranov’s Lie complexes [13], Hacking, Keel and Tevelev’s very stable pairs [11] and the author’s tropical linear spaces when they are realizable in characteristic zero [25]. Namely, in characteristic zero, a Lie complex or the underlying d − 1 dimensional scheme of a very stable pair can have at (n−i−1)! (d−i)!(n−d−i)!(i−1)! most strata of dimensions n − i and d − i respectively and a tropical linear space realizable in characteristic zero can have at most this many idimensional bounded faces.
Graph Decompositions and Factorizing Permutations
 Discrete Mathematics and Theoretical Computer Science
, 1997
"... A factorizing permutation of a given undirected graph is simply a permutation of the vertices in which all decomposition sets appear to be factors. Such a concept seems to play a central role in recent papers dealing with graph decomposition. It is applied here for modular decomposition and we propo ..."
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Cited by 11 (6 self)
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A factorizing permutation of a given undirected graph is simply a permutation of the vertices in which all decomposition sets appear to be factors. Such a concept seems to play a central role in recent papers dealing with graph decomposition. It is applied here for modular decomposition and we propose a linear algorithm that computes the whole decomposition tree when a factorizing permutation is provided. This algorithm can be seen as a common generalization of Ma and Hsu [9, 8] for modular decomposition of chordal graphs and Habib, Huchard and Spinrad [7] for inheritance graphs decomposition. It also suggests many new decomposition algorithms for various notions of graph decompositions.
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.
The structure of locally finite twoconnected graphs
 Electron J. Combin
, 1995
"... We expand on Tutte's theory of 3blocks for 2connected graphs, generalizing it to apply to infinite, locally finite graphs, and giving necessary and sufficient conditions for a labeled tree to be the 3block tree of a 2connected graph. ..."
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Cited by 8 (3 self)
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We expand on Tutte's theory of 3blocks for 2connected graphs, generalizing it to apply to infinite, locally finite graphs, and giving necessary and sufficient conditions for a labeled tree to be the 3block tree of a 2connected graph.