Connected matroids with a small circumference

Lemos and Oxley proved that if M is a connected matroid with |E(M)|⩾3r(M), then M has a circuit C such that M⧹C is connected. In this paper, we shall improve this result proving that for a simple and connected matroid M, if r(M)⩾7 and |E(M)|⩾3r(M)−3, then M has a circuit C such that M⧹C is connected. To prove this result, we shall construct all the connected matroids having circumference at most five, with the exception of those which are 3-connected and have rank five.     

Least and most colored bases

Consider a matroid M=(E,B), where B denotes the family of bases of M, and assign a color c(e) to every element e∈E (the same color can go to more than one element). The palette of a subset F of E, denoted by c(F), is the image of F under c. Assume also that colors have prices (in the form of a function π(?), where ? is the label of a color), and define the chromatic price as: π(F)=∑_?∈c(F)π(?). We consider the following problem: find a base B∈B such that π(B) is minimum. We show that the greedy algorithm delivers a lnr(M)-approximation of the unknown optimal value, where r(M) is the rank of matroid M. By means of a reduction from SETCOVER, we prove that the lnr(M) ratio cannot be further improved, even in the special case of partition matroids, unless . The results apply to the special case where M is a graphic matroid and where the prices π(?) are restricted to be all equal. This special case was previously known as the minimum label spanning tree (MLST) problem. For the MLST, our results improve over the ln(n-1)+1 ratio achieved by Wan, Chen and Xu in 2002. Inspired by the generality of our results, we study the approximability of coloring problems with different objective function π(F), where F is a common independent set on matroids M₁,…,M_k and, more generally, to independent systems characterized by the k-for-1 property.     

Connected hyperplanes in binary matroids

For a 3-connected binary matroid M, let dim_A(M) be the dimension of the subspace of the cocycle space spanned by the non-separating cocircuits of M avoiding A, where A⊆E(M). When A=∅, Bixby and Cunningham, in 1979, showed that dim_A(M)=r(M). In 2004, when |A|=1, Lemos proved that dim_A(M)=r(M)-1. In this paper, we characterize the 3-connected binary matroids having a pair of elements that meets every non-separating cocircuit. Using this result, we show that 2dim_A(M)?r(M)-3, when M is regular and |A|=2. For |A|=3, we exhibit a family of cographic matroids with a 3-element set intersecting every non-separating cocircuit. We also construct the matroids that attains McNulty and Wu’s bound for the number of non-separating cocircuits of a simple and cosimple connected binary matroid.     

Vector representable matroids of given rank with given automorphism group

A simple way of associating a matroid of prescribed rank with a graph is shown. The matroids so constructed are representable over any sufficiency large field. Their use is demonstrated by the following result: Given an integer k?3 and a function G associating a group with each subset of a set S, there is a matroid M(E), representable over any sufficiently large field, such that E ? S, and for any T ?/ S, the rank of M/Tis k, and the automorphine group of MT is isomorphic to G(T).     

A chain theorem for internally 4-connected binary matroids

Let M be a matroid. When M is 3-connected, Tutte's Wheels-and-Whirls Theorem proves that M has a 3-connected proper minor N with |E(M)−E(N)|=1 unless M is a wheel or a whirl. This paper establishes a corresponding result for internally 4-connected binary matroids. In particular, we prove that if M is such a matroid, then M has an internally 4-connected proper minor N with |E(M)−E(N)|?3 unless M or its dual is the cycle matroid of a planar or Möbius quartic ladder, or a 16-element variant of such a planar ladder.     

On Maximum-sized k-Regular Matroids

 Let k be an integer exceeding one. The class of k-regular matroids is a generalization of the classes of regular and near-regular matroids. A simple rank-r regular matroid has the maximum number of points if and only if it is isomorphic to M(K _r+1), the cycle matroid of the complete graph on r+1 vertices. A simple rank-r near-regular matroid has the maximum number of points if and only if it is isomorphic to the simplification of , that is, the simplification of the matroid obtained, geometrically, by freely adding a point to a 3-point line of M(K _r+2) and then contracting this point. This paper determines the maximum number of points that a simple rank-r k-regular matroid can have and determines all such matroids having this number. With one exception, there is exactly one such matroid. This matroid is isomorphic to the simplification of , that is, the simplification of the matroid obtained, geometrically, by freely adding k independent points to a flat of M(K _r+k+1) isomorphic to M(K _k+2) and then contracting each of these points. Revised: July 27, 1998     

Spaces of Riemannian metrics on open manifolds

We define for the set M of metrics on an open manifold M ⁿ suitable uniform structures, obtain completed spaces ^b,m M or M ^r (I, B _k ), respectively and calculate for each component of M ^r (I, B _k ) the infinitedimensional geometry. In particular, we show that the sectional curvature is non positive.     

On the Cauchy problem for some abstract nonlinear differential equations

In the present paper, we study the Cauchy problem in a Banach spaceE for an abstract nonlinear differential equation of form $$\frac{{d^2 u}}{{dt^2 }} = - A\frac{{du}}{{dt}} + B(t)u + f(t,W)$$ whereW = (A ₁(t)u,A ₂(t)u,?,A _?(t)u), (A _i (t),i = 1, 2, ?,?), (B(t),t ∈I = [0,b]) are families of closed operators defined on dense sets inE intoE, f is a given abstract nonlinear function onI ×E ^? intoE and ?A is a closed linear operator defined on dense set inE intoE, which generates a semi-group. Further, the existence and uniqueness of the solution of the considered Cauchy problem is studied for a wide class of the families (A _i(t),i = 1, 2, ?,?), (B(t),t ∈I). An application and some properties are also given for the theory of partial diferential equations.     

On 3-connected minors of 3-connected matroids and graphs

Whittle proved, for k=1,2, that if N is a 3-connected minor of a 3-connected matroid M, satisfying r(M)−r(N)≥k, then there is a k-independent set I of M such that, for every x∈I, si(M/x) is a 3-connected matroid with an N-minor. In this paper, we establish this result for k=3. It is already known that it cannot be extended to greater values of k. But, here we also show that, in the graphic case, with the extra assumption that r(M)−r(N)≥6, we can guarantee the existence of a 4-independent set of M with such a property. Moreover, in the binary case, we show that if r(M)−r(N)≥5, then M has such a 4-independent set or M has a triangle T meeting 3 triads and such that M/T is a 3-connected matroid with an N-minor.     

A Splitter Theorem Relative to a Fixed Basis

A standard matrix representation of a matroid M represents M relative to a fixed basis B, where contracting elements of B and deleting elements of E(M)–B correspond to removing rows and columns of the matrix, while retaining standard form. If M is 3-connected and N is a 3-connected minor of M, it is often desirable to perform such a removal while maintaining both 3-connectivity and the presence of an N-minor. We prove that, subject to a mild essential restriction, when M has no 4-element fans with a specific labelling relative to B, there are at least two distinct elements, s ₁ and s ₂, such that for each i ∈ {1, 2}, si(M/s _i ) is 3-connected and has an N-minor when s _i ∈ B, and co(M\s _i ) is 3-connected and has an N-minor when s _i ∈ E(M)–B. We also show that if M has precisely two such elements and P is the set of elements that, when removed in the appropriate way, retain the N-minor, then (P, E(M)–P) is a sequential 3-separation.     

Construction of matroidal families by partly closed sets

A matroidal family of graphs is a set $\text{M}$≠Ø of connected finite graphs such that for every finite graph G the edge sets of those subgraphs of G which are isomorphic to some element of $\text{M}$ are the circuits of a matroid on the edge set of G. In [9], Schmidt shows that, for n?0, ?2n<r?1, n, $\text{r∈}\text{Z}$, the set $\text{M}\text{(n, r)={G∣G}$ is a graph with β(G)=nα(G)+r and α(G )>, and is minimal with this property (with respect to the relation ?))} is a matroidal family of graphs. He also describes a method to construct new matroidal families of graphs by means of so-called partly closed sets. In this paper, an extension of this construction is given. By means of s-partly closed subsets of $\text{M}\text{(n, r)}$, s?r, we are able to give sufficient and necessary conditions for a subset $\text{P}\text{(n, r)}$ of $\text{M}\text{(n, r)}$ to yield a matroidal family of graphs when joined with the set $\text{I}\text{(n, s)}$ of all graphs $\text{G∈}\text{M}\text{(n, s)}$ which satisfy: If $\text{H∈}\text{P}\text{(n, r)}$, then H?G. In particular, it is shown that $\text{M}\text{(n, r)}$ is not a matroidal family of graphs for r?2. Furthermore, for n?0, 1?2n<r, n, $\text{r∈}\text{Z}$, the set of bipartite elements of $\text{M}\text{(n, r)}$ can be used to construct new matroidal families of graphs if and only if s?min(n+r, 1).     

Testing membership in matroid polyhedra

Given a matroid M on E and a nonnegative real vector x=(x_j:j∈E), a fundamental problem is to determine whether x is in the convex hull P of (incidence vectors of) independent sets of M. An algorithm is described for solving this problem for which the amount of computation is bounded by a polynomial in |E|, independently of x, allowing as steps tests of independence in M and additions, subtractions, and comparisons of numbers. In case x ∈ P, the algorithm finds an explicit representation for x which has additional nice properties; in case x ? P it finds a most-violated inequality of the system defining P. The same technique is applied to the problem of finding a maximum component-sum vector in the intersection of two matroid polyhedra and a box.     

The structure of the 3-separations of 3-connected matroids

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. When M is 2-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 non-trivial 3-separations of M.     

Matroid optimization with the interleaving of two ordered sets

We study the structure of the minimum weight base of a matroid M = (E, I) the order of whose element set E is determined by the interleaving of two ordered subsets of E, R and W. The results imply an interesting application in economics, and are useful for the rapid recomputation of the minimum weight base when the order of E is successively modified by changing the interleaving of R and W. As a special case of the main result, the following parametric problem is efficiently solved: For M = (E, I) a matroid with weighted element set E, and R a subset of E, find for all feasible values of q, the minimum weight base of M containing exactly q elements of R. This parametric problem is a weighted matroid intersection problem and hence can be solved by known matroid intersection algorithms. The approach in this paper is different, and vastly improves the efficiency of the solution, as well as determining structural information about the bases.     

A common property of R(E, F) and B(ℝ n , ℝ m ) and a new method for seeking a path to connect two operators

Given two Banach spaces E, F, let B(E, F) be the set of all bounded linear operators from E into F, and R(E, F) the set of all operators in B(E, F) with finite rank. It is well-known that B(? ⁿ ) is a Banach space as well as an algebra, while B(? ⁿ , ? ^m ) for m ≠ n, is a Banach space but not an algebra; meanwhile, it is clear that R(E, F) is neither a Banach space nor an algebra. However, in this paper, it is proved that all of them have a common property in geometry and topology, i.e., they are all a union of mutual disjoint path-connected and smooth submanifolds (or hypersurfaces). Let Σ _r be the set of all operators of finite rank r in B(E, F) (or B(? ⁿ , ? ^m )). In fact, we have that 1) suppose Σ _r ∈ B(? ⁿ , ? ^m ), and then Σ _r is a smooth and path-connected submanifold of B(? ⁿ , ? ^m ) and dimΣ _r = (n + m)r ? r ², for each r ∈ [0, min{n,m}; if m ≠ n, the same conclusion for Σ _r and its dimension is valid for each r ∈ [0, min{n, m}]; 2) suppose Σ _r ∈ B(E, F), and dimF = ∞, and then Σ _r is a smooth and path-connected submanifold of B(E, F) with the tangent space T _A Σ _r = {B ∈ B(E, F): BN(A) ? R(A)} at each A ∈ Σ _r for 0 ? r ? ∞. The routine methods for seeking a path to connect two operators can hardly apply here. A new method and some fundamental theorems are introduced in this paper, which is development of elementary transformation of matrices in B(? ⁿ ), and more adapted and simple than the elementary transformation method. In addition to tensor analysis and application of Thom’s famous result for transversility, these will benefit the study of infinite geometry.     

The convex weighting of a graph and an alternative definition of a matroid

In 1976, R.N. Burns and C.E. Haff gave an algorithm for finding the kth-best spanning tree of an edge-weighted graph as well as the kth-best base of an element-weighted matroid. In this paper, after introducing the concept of a convex weight function defined on the vertex set of a connected graph, the following result is proved: Let H = (S, $\text{I}$) be an independence system, where $\text{I}$ is the set of independent subsets of H, such that all the maximal independent subsets of H are of the same cardinality. Then a necessary and sufficient condition for H to be a matroid is that, for any weight function W defined on S, the algorithm of Burns and Haff gives a labelling of the family of maximal sets in $\text{I}$ as B₁, B₂, …, B_n such that W(B₁) ? W(B₂) ? ··· ? W(B_n).     

The connectivity and minimum degree of circuit graphs of matroids

Let G be the circuit graph of any connected matroid M with minimum degree 5（G）. It is proved that its connectivity κ（G） ≥2｜E（M） - B（M）｜ - 2. Therefore 5（G） ≥ 2｜E（M） - B（M）｜ - 2 and this bound is the best possible in some sense.     

Maximum modulus sets in pseudoconvex boundaries

LedD be a strictly pseudoconvex domain in ? ⁿ withC ^∞ boundary. We denote byA ^∞(D) the set of holomorphic functions inD that have aC ^∞ extension to \(\bar D\) . A closed subsetE of ?D is locally a maximum modulus set forA ^∞(D) if for everyp∈E there exists a neighborhoodU ofp andf∈A ^∞(D∩U) such that |f|=1 onE∩U and |f|<1 on \(\bar D \cap U\backslash E\) . A submanifoldM of ?D is an interpolation manifold ifT _p (M)?T _p ^c (?D) for everyp∈M, whereT _p ^c (?D) is the maximal complex subspace of the tangent spaceT _p (?D). We prove that a local maximum modulus set forA ^∞ (D) is locally contained in totally realn-dimensional submanifolds of ?D that admit a unique foliation by (n?1)-dimensional interpolation submanifolds. LetD =D ₁ x ... xD _r ? ? ⁿ whereD _i is a strictly pseudoconvex domain withC ^∞ boundary in ? ⁿ _i ,i=1,…,r. A submanifoldM of ?D ₁×…×?D _r verifies the cone condition if \(II_p (T_p (M)) \cap \bar C[Jn_1 (p),...,Jn_r (p)] = \{ 0\} \) for everyp∈M, wheren _i (p) is the outer normal toD _i atp, J is the complex structure of ? ⁿ , \(\bar C[Jn_1 (p),...,Jn_r (p)]\) is the closed positive cone of the real spaceV _p generated byJ _{n 1}(p),…,J _{n r}(p), and II _p is the orthogonal projection ofT _p (?D) onV _p . We prove that a closed subsetE of ?D ₁×…×?D _r which is locally a maximum modulus set forA ^∞(D) is locally contained inn-dimensional totally real submanifolds of ?D ₁×…×?D _r that admit a foliation by (n?1)-dimensional submanifolds such that each leaf verifies the cone condition at every point ofE. A characterization of the local peak subsets of ?D ₁×…×?D _r is also given.     

Blockers and transversals in some subclasses of bipartite graphs: When caterpillars are dancing on a grid

Given an undirected graph G=(V,E) with matching number ν(G), a d-blocker is a subset of edges B such that ν((V,E?B))≤ν(G)−d and a d-transversal T is a subset of edges such that every maximum matching M has |M∩T|≥d. While the associated decision problem is NP-complete in bipartite graphs we show how to construct efficiently minimum d-transversals and minimum d-blockers in the special cases where G is a grid graph or a tree.     

Disjoint Kr-minors in large graphs with given average degree

It is proved that there are functions f(r) and N(r,s) such that for every positive integer r, s, each graph G with average degree d(G)=2|E(G)|/|V(G)|≥f(r), and with at least N(r,s) vertices has a minor isomorphic to K_r,s or to the union of s disjoint copies of K_r.