The next high-priority phase of human genomics will involve the development of a full Haplotype Map of the human genome [12]. It will be used in large-scale screens of populations to associate specific haplotypes with specific complex genetic-influenced 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

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 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 well-known 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 -stable-set separators; polynomial-time 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 polynomial-time 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 . E-mail: tomas@theory.stanford.edu. y School of Computing Science, Simon Fraser University, Burnaby, B.C., Canada, V5A1S6. E-mail: 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, 21945-970, Brasil. E-mail: sula@cos.ufrj.br. Supported by CNPq and PRONEX 107/97. x Department of Computer Science, Stanford University, CA 94305-9045. E-mail: rajeev@cs.stanford.edu. Supported by an ARO MURI Grant DAAH04--96--1...

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 delta-matroids and matroids. Copyright (C) by the Society for Industrial and Applied Mathematics, in SIAM Journal on Discrete Mathematics, 8 (1995) pp. 17--32. 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...

A polynomial P in n complex variables is said to have the “half-plane property” (or Hurwitz property) if it is nonvanishing whenever all the variables lie in the open right half-plane. 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 r-uniform set system. We prove that the support (set of nonzero coefficients) of a homogeneous multiaffine polynomial with the half-plane 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 half-plane property? Not all matroids have the half-plane property, but we find large classes that do: all sixth-root-of-unity matroids, and a subclass of transversal (or cotransversal) matroids that we call “nice”. Furthermore, the class of matroids with the half-plane property is closed under minors, duality, direct sums, 2-sums, series and parallel connection, full-rank 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 half-plane 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,

Let G(d,n) denote the Grassmannian of d-planes 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 T-orbit through x. Then the class of the structure sheaf of Tx in the K-theory 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 i-dimensional bounded faces.

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.

This paper gives an O(n²) incremental algorithm for computing the modular decomposition of 2-structure [1, 2]. A 2-structure is a type of edge-colored graph, and its modular decomposition is also known as the prime tree family. Modular decomposition of 2-structures arises in the study of relational systems. The modular decomposition of undirected graphs and digraphs is a special case, and has applications in a number of combinatorial optimization problems. This algorithm generalizes elements of a previous O(n²) algorithm of Muller and Spinrad [3] for the decomposition of undirected graphs. However, Muller and Spinrad's algorithm employs a sophisticated data structure that impedes its generalization to digraphs and 2-structures, and limits its practical use. We replace this data structure with a scheme that labels each edge with at most one node, thereby obtaining an algorithm that is both practical and general to 2-structures.

Let M be a matroid. When M is 2-connected, Cunningham and Edmonds gave a tree decomposition of M that displays all of its 2-separations. This result was extended by Oxley, Semple, and Whittle, who showed that, when M is 3-connected, there is a corresponding tree decomposition that displays all non-trivial 3-separations of M up to a certain natural equivalence. This equivalence is based on the notion of the full closure fcl(Y)ofasetY in M, which is obtained by beginning with Y and alternately applying the closure operators of M and M ∗ until no new elements can be added. Two 3-separations (Y1,Y2) and(Z1,Z2) are equivalent if {fcl(Y1), fcl(Y2)} = {fcl(Z1), fcl(Z2)}. The purpose of this paper is to identify all the structures in M that lead to two 3-separations being equivalent and to describe the precise role these structures have in determining this equivalence.

We expand on Tutte's theory of 3-blocks for 2-connected graphs, generalizing it to apply to infinite, locally finite graphs, and giving necessary and sufficient conditions for a labeled tree to be the 3-block tree of a 2-connected graph.