Compositions for matroids with the Fulkerson property (original) (raw)
The positive circuits of oriented matroids with the packing property or idealness
Electronic Notes in Discrete Mathematics, 2010
The class of clutters of the positive circuits of oriented matroids is an important class which generalizes both the dicut clutters and the dicycle clutters of directed graphs, together. In this article, we will show that the clutter of the positive circuits of an oriented matroid whose co-rank is at most 4 has the packing property if and only if it has none of the six minimally non-packing minors J 3 , C 2 3 , C 3 5 , Q 6 , Q 6 ⊗ 1 and Q 6 ⊗ {1, 2}. By using this, we will also prove that the clutter of the positive circuits of an oriented matroid whose co-rank is at most 4 is ideal if and only if it has none of the three minimally non-packing minors J 3 , C 2 3 , and C 3 5 .
Some new subclasses of graphic matroids, related to the union operation
2015
There is a conjecture that if the union (also called sum) of graphic matroids is not graphic then it is nonbinary. Some special cases have been proved only, for example if several copies of the same graphic matroid are given. If there are two matroids and the first one can either be represented by a graph with two points, or is the direct sum of a circuit and some loops, then a necessary and sufficient condition is known for the other matroid to ensure the graphicity of the union and the above conjecture holds for these cases. We have proved the sufficiency of this condition for the graphicity of the union of two arbitrary graphic matroids. Then we present a weaker necessary condition which is of similar character. Finally we suggest a more general framework of the study of such questions by introducing matroid classes formed by those graphic (or arbitrary) matroids whose union with any graphic (or arbitrary) matroid is graphic (or either graphic or nonbinary).
Circuit and cocircuit partitions of binary matroids
Czechoslovak Mathematical Journal, 2006
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Uniform partitioning to bases in a matroid Zsolt Fekete ? and Jácint Szabó ? ?
2005
We say that a matroid M is k-uniform if – provided that it is the disjoint union of its two bases – for any given k-element subpartition P of its ground set, M can be partitioned into two disjoint bases B1, B2 such that ||B1 ∩ P | − |B2 ∩ P || ≤ 1 for all P ∈ P. The circuit matroid of an undirected graph G is called k-star-uniform if the above holds for all k-element subpartitions containing stars of independent vertices of G. In this paper we prove that the circuit matroids are 1-uniform and 3-star-uniform but not necessarily 2-uniform and 4-star-uniform.
On the circuit-spectrum of binary matroids
2011
Murty, in 1971, characterized the connected binary matroids with all circuits having the same size. We characterize the connected binary matroids with circuits of two different sizes, where the largest size is odd. As a consequence of this result we obtain both Murty's result and other results on binary matroids with circuits of only two sizes. We also show that it will be difficult to complete the general case of this problem.
Boolean designs and self-dual matroids
Linear Algebra and its Applications, 1975
In this paper we consider a variety of questions in the context of Boolean designs. For example, ErdGs asked: How many subsets of an n-set can be found so that pairwise their intersections are all even (odd)? E. Berlekamp [2] and the author both answered this question; the answer is approximately 2[b"'. Another question which can be formulated in terms of Boolean designs was asked by J. A. Bondy and D. J. A. Welsh [l]. For what values of d can one find a connected binary matroid of rank d which is identically self-dual? We prove that such matroids exist for all d except 2, 3, and 5. The paper ends with a discussion of more general modular designs and with constructions of some identically self-dual matroids representable over the field of three elements. 0. INTRODUCTION Let U be a finite set. We denote the collection of all subsets of U by 9 (U), and we let 5?r (U) denote the collection of all r-subsets of U. A triple (U,f, 3) is called a system of blocks if Ci3 is also a finite set and f maps ?i; into C? (U). The elements of U are called the vertices of the system; the elements of '33 are called the blocks of the system. If f is an injection, (C',f, %I) is called a system of sets. In this case we often identify % with its image under f and simply write (U, 3), where Ci3 c 9 (U). We will be primarily interested in systems of sets; however, we are forced to discuss the more general situation because systems of sets are not closed under duality. Let
A matroid generalization of a theorem of Mendelsohn and Dulmage
Discrete Mathematics, 1973
A matrwd generalization is given to a theorem of Mendrlsohn and Dulmape mnwning assignments in bipartite graphs. The generalized theorem hat applications ilr optimiralkm theory and provides a simple proof of a theorem of Nash-Williams.
The operation of matroid union was introduced by Nash-Williams in 1966. A matroid is indecomposable if it cannot be written in the form M = M1 V M2, where r(M1),r(M2) > 0. In 1971 Welsh posed the problem of characterizing indecomposable matroids, this problem has turned out to be extremely difficult. As a partial solution towards its progress, Cunningham characterized binary indecomposable matroids in 1977. In this thesis we present numerous results in topics of matroid union. Those include a link between matroid connectivity and matroid union, such as the implication of having a 2-separation in the matroid union, and under what conditions is the union 3-connected. We also identify which elements in binary and ternary matroids are non-fixed. Then we create a link between having non-fixed elements in binary and ternary matroids and the decomposability of such matroids, and the effect of removing non-fixed elements from binary and ternary matroids. Moreover, we show results concern...
A short proof of the Truemper-Tseng theorem on max-flow min-cut matroids
Linear Algebra and its Applications, 1989
Seymour has characterized the matroids satisfying the integral max-flow min-cut property with respect to a fixed element (Se77b). Truemper and Tseng (TsTr86) subsequently proved a decomposition theorem for this class, similar in spirit to Wagner's characterization of the graphs containing no K 5 minor (Wa37) and Seymour's characterization of the regular (totally unimodular) matroids (Se80). The purpose of this paper is to give a short, self-contained 1 exposition of the Truemper-Tseng result. 2 Max-Flow Min-Cut Matroids Throughout this paper M denotes a matroid on a finite set E. Fix l E E, and let A be the {O, 1}-matrix with columns indexed on elements e E E-l (braces being omitted since { l} is a singleton) and rows indexed on circuits C of M containing l, such that the (C, e) entry is 1 iff e E C. In the case when M is graphic, the rows of A correspond to paths joining the end-vertices of l. Let C* be the family of cocircuits of M containing l. We say that M is 1-MFMC, that is, has the (integral) max-flow min-cut property with respect to l, if for every choice of nonnegative integral vector w defined on E, min w(C*-l) = max {yTl : yT A~ wT, y ~ 0 and integral}, c•ec•
Matroid Partition Property and the Secretary Problem
ArXiv, 2021
A matroid M on a set E of elements has the α-partition property, for some α > 0, if it is possible to (randomly) construct a partition matroid P on (a subset of) elements of M such that every independent set of P is independent in M and for any weight function w : E → R≥0, the expected value of the optimum of the matroid secretary problem on P is at least an α-fraction of the optimum on M. We show that the complete binary matroid, Bd on Fd2 does not satisfy the α-partition property for any constant α > 0 (independent of d). Furthermore, we refute a recent conjecture of [BSY21] by showing the same matroid is 2/d-colorable but cannot be reduced to an α2/d-colorable partition matroid for any α that is sublinear in d. dornaa@cs.washington.edu. Research supported by NSF grant CCF-1907845 and Air Force Office of Scientific Research grant FA9550-20-1-0212. karlin@cs.washington.edu. Research supported by Air Force Office of Scientific Research grant FA9550-20-10212 and NSF grant CCF-1...
A note on packing spanning trees in graphs and bases in matroids
Cornell University - arXiv, 2012
We consider the class of graphs for which the edge connectivity is equal to the maximum number of edge-disjoint spanning trees, and the natural generalization to matroids, where the cogirth is equal to the number of disjoint bases. We provide descriptions of such graphs and matroids, showing that such a graph (or matroid) has a unique decomposition. In the case of graphs, our results are relevant for certain communication protocols.
The graphicity of the union of graphic matroids
European Journal of Combinatorics, 2015
There is a conjecture that if the union (also called sum) of graphic matroids is not graphic then it is nonbinary [7]. Some special cases have been proved only, for example if several copies of the same graphic matroid are given. If there are two matroids and the first one can either be represented by a graph with two points, or is the direct sum of a circuit and some loops, then a necessary and sufficient condition is given for the other matroid to ensure the graphicity of the union. These conditions can be checked in polynomial time. The proofs imply that the above conjecture holds for these cases.
Algorithms for Enumerating Circuits in Matroids
Lecture Notes in Computer Science, 2003
We present an incremental polynomial-time algorithm for enumerating all circuits of a matroid or, more generally, all minimal spanning sets for a flat. This result implies, in particular, that for a given infeasible system of linear equations, all its maximal feasible subsystems, as well as all minimal infeasible subsystems, can be enumerated in incremental polynomial time. We also show the NP-hardness of several related enumeration problems.
A Note on Maxflow-Mincut and Homomorphic Equivalence in Matroids
Journal of Algebraic Combinatorics - J ALGEBR COMB, 2000
Graph homomorphisms are used to study good characterizations for coloring problems Trans. Amer. Math. Soc. 384 (1996), 1281–1297; Discrete Math.22 (1978), 287–300). Particularly, the following concept arises in this context: A pair of graphs (A, B) is called a homomorphism duality if for any graph G either there exists a homomorphism s : A ? G or there exists a homomorphism t : G ? B but not both. In this paper we show that maxflow-mincut duality for matroids can be put into this framework using strong maps as homomorphisms. More precisely, we show that, if Ck denotes the circuit of length k + 1, the pairs (Ck, Ck + 1) are the only homomorphism dualities in the class of duals of matroids with the strong integer maxflow-mincut property (Jour. Comb. Theor. Ser.B23 (1977), 189–222). Furthermore, we prove that for general matroids there is only a trivial homomorphism duality.
On Critical Circuits in k-Connected Matroids
Graphs and Combinatorics, 2018
We show that, for every integer k ≥ 4, if M is a k-connected matroid and C is a circuit of M such that for every e ∈ C, M\e is not k-connected, then C meets a cocircuit of size at most 2k − 3; furthermore, if M is binary and k ≥ 5, then C meets a cocircuit of size at most 2k − 4. It follows from our results and a result of Reid et al [5] that every minimally k-connected matroid has a cocircuit of size at most 2k − 3, and every minimally k-connected binary matroid has a cocircuit of size at most 2k − 4.