Questions related to the field of Combinatorics called Matroid Theory. Relevant topics include matroids in Combinatorial Optimization, Lattice Theory, Algebraic Geometry, Polyhedral Theory, Rigidity, and Algorithms. For questions about Oriented Matroids, the oriented-matroids tag may be used.

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Matroid Representation of the Antichains of a Poset

Introduction I am studying a problem in which the antichains of a poset are of key importance. They are naturally geometrically embedded as vectors in the space $\mathbb{R}^P$, where $P$ is the poset,...
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1answer
36 views

Augmention property of matroid along perfect matching

Let M be a matroid of rank k, B a base, X a set of rank rank(X) < k, and P a perfect matching of the complete bipartite graph (X, B). Is it true that there exists an edge (x, b) of P augmenting X (...
3
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1answer
102 views

Base decomposition of matroids

I want to find a generalization of the idea that, in a graphic matroid, every base can be decomposed on the stars (edges adjacent to a vertex). For example one could say that a matroid $M$ of rank $k$...
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0answers
62 views

Which matroids have not unique unimodular representation?

Matroid $M$ is represented by real vectors, and we know that any base of $M$ generates the same lattice (this is called unimodular representation, I guess.) If we change the sign of any vector, we ...
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117 views

Determine existence of matroid with some barrier given

Let $E$ be a finite ground set. Let $\mathcal{L}$ (as lower barrier) and $\mathcal{U}$ (upper) be subsets of $2^E$. How can we determine whether there is some matroid $\mathcal{M}=(E,\mathcal{I})$ ...
7
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1answer
69 views

minimum number of bases of a matroid, that comes from a convex polytope

Given a d-dimensional polytope P with n points, then what is the minimum number of simplices that are spanned by vertices of P? This question led my research to matroids and so my question is: what is ...
3
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0answers
38 views

Weighted matching and spanning set in $2$-polymatroids

A $2$-polymatroid $P$ is a pair $(S, f)$, where $S$ is a finite set and $f: 2^{S} \rightarrow \mathbb{Z}$ is a function satisfying the following: $f(\varnothing) = 0$; $f(X) \leq f(Y)$, for any $X ...
3
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1answer
208 views

Is there a version of Robertson-Seymour's graph minor theorem known to apply to signed graphs?

Here a signed graph is one where each edge is signed either odd or even. A cycle is odd or even according to the sum of the signs of its edges. For a given signed graph, a resigning may be performed ...
6
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1answer
105 views

New base of matroid from olds

Let M be a matroid of rank 3 and $E_1, E_2, E_3$ 3 basis of M. Let $e_{i,j}$ be the ith element of base $E_j$. Is it true that you can always find a permutation s: {1,2,3} -> {1,2,3} such that $e_{s(1)...
5
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1answer
127 views

Matroids similar to the cycle matroid

Let $G=(V,E)$ be a graph (loops and multiple edges are permitted). Three following systems of dependent sets in $E$ define matroids: 1) Set $A\subset E$ is dependent if $A$ contains cycle. This is a ...
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1answer
192 views

Number of bases of a matroid

I would like to know the minimum number of bases of a matroid of rank $k$ and $n$ elements, knowing that each singleton is independent. At least for small ranks.
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1answer
58 views

Do the support sets of subspaces give the representable matroids?

Fact:   Start with $V$ a subspace of $\mathbb R^n$. Take the set of all supports of vectors in $V$. Throw out $\emptyset$. You now have the dependent sets of some matroid. Not sure you ...
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0answers
33 views

Non-uniform matroids as the matroid sum of uniform matroids

Can all non-uniform matroids be written as the direct sum / matroid sum of uniform matroids? If so, What happens to the matrices representing the uniform matroids? If the non-uniform matroid is ...
3
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0answers
135 views

A Result of Anders Bjorner: Matchings in countably infinite geometric lattices of finite height

Let $L$ be a countably infinite geometric lattice of finite height $r\ge3$. (A geometric lattice of height $r$ is an atomistic semimodular lattice such that every maximal chain has $r+1$ elements.) ...
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1answer
101 views

Is the union of strongly base-orderable matroids strongly base-orderable?

A matroid is said to be strongly base-orderable if for any two bases $B_1,B_2$ there is a bijection $f:B_1 \to B_2$ such that for any $S\subseteq B_1$ set $(B_1 \setminus S) \cup f(S)$ is also a base. ...
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0answers
167 views

Determining strong base-orderability of a matroid

A matroid is said to be strongly base-orderable when for any two bases $B_1,B_2$ there exists a bijection $f:B_1 \mapsto B_2$ such that for any $X\subseteq B_1$ set $B_1 - X+ f(X)$ is also a base. ...
3
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0answers
78 views

Non-representable irreducible matroid of rank at least 5?

Can anyone tell me an example of a matroid of rank 5 or higher which is not a product of two lower rank matroids and is not the independence matroid of a finite set of vectors in a vector space over ...
4
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2answers
202 views

Matroids relaxations of a given matroid

Let $\mathcal{M}$ be a rank-$d$ matroid on $[n]$. Say a matroid $\mathcal{N}$ is a relaxation of $\mathcal{M}$ if $\mathrm{rank}(\mathcal{N})=d$, $\mathrm{groundset}(\mathcal{N})=[n]$, and every ...
5
votes
2answers
299 views

Decomposing polyhedral cones into “direct sums” and a polynomial

This question consists of two parts. I'm not breaking it up into two separate ones because posing the second question would essentially require me two rewrite the first one. Also, to some extent, the ...
5
votes
1answer
269 views

Small remarkable matroids

Working on phased matroids (a generalization of oriented matroid to the complex case) I've found an interesting formula for computing the inner Tutte group (and, hence, all the Tutte groups) ...
5
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1answer
294 views

Vanishing patterns of minors of matrix

Let $M$ be a $m\times n$ matrix with entries in, say $\mathbb{C}$; assume $n\leq m$. Denote by $I\subseteq\{1,2,\ldots, n\}$ a subset of the columns of M. I am interested in positive results to the ...
5
votes
2answers
78 views

The lattice of covectors of an oriented matroid

Let $M$ be an oriented matroid on the ground set $E$, and let $L(M)$ be its ranked poset of covectors. By definition, $L(M)$ is a sub-poset of the poset $\{0, \pm 1\}^E$, ordered by putting $0 < \...
5
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1answer
332 views

Does this matroid have a name?

Sorry if this question is a dumb one. I used a special family of matroids in my research. One of them, of rank 3, can be represented by the following matrix over $\mathbb F_5$ or over $\mathbb R$: $\...
2
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1answer
279 views

Positroids and Totally Nonnegative Complex Grassmanian

Recently I begin working on matroids, in particular to a generalization of oriented matroids to the complex case. I found on arxiv the following interesting articles: 1)Alexander Postnikov: Total ...
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0answers
78 views

Characterizing bases of 0-1-vectors in $\mathbb{R}^n$ in terms of their partial order

Given $n \in \mathbb{N}$ and $V \subseteq \{0,1\}^n$, can properties of $V$ with respect to the vector space $\mathbb{R}^n$ (not $\mathbb{Z}_2^n$), $V$ is linearly independent $\dim \mathrm{span} V ...
2
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1answer
206 views

Looking for a canonical (matroid polytope) subdivision of the hypersimplex

A matroid polytope is the convex hull of the indicator vectors of the bases of a matroid, and a matroid polytope subdivision (MPS) is a polyhedral subdivision of a matroid polytope whose cells are ...
2
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2answers
158 views

Which graphs generate a matroidal independence complex?

The independence complex $I(G)$ of a graph $G=(V,E)$ has as point set the vertex set $V$ and as simplices the independent sets of $G$. Now, if $G$ is a well-covered graph (where all maximal ...
4
votes
2answers
260 views

matroids axioms and independence system

A finite matroid $M$ is a pair $(E,I)$ where $E$ is a finite set and $I$ is a family of independent set with the following properties: 1) There is at least an independent system 2) Every subset of ...
0
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1answer
106 views

Basis of Cycle Subspace of a Graph

Let $G$ be a $2$-connected graph and for $e \in E(G)$ denote by $\mathcal{C_e}$ the set of all cycles(circuits) of $G$ containing the edge $e$. For what set of edges does $\mathcal{C_e}$ contain a ...
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1answer
389 views

Fundamental Cycles of a graphs

For a $2$-edge-connected simple graph $G$ and a tree $T$ of $G$, let $C_e$ be the unique cycle in $T + e$, $e \in E(G) - E(T)$. Define the set $\mathcal{C}(T) = \{C_e | e \in E(G) - E(T)\}$. Now ...
3
votes
2answers
249 views

Does this graph have a name?

Let $G$ be a connected graph on $n$ vertices and $\mathcal{T}$ be the set of all spanning trees of $G$. Consider the graph whose vertices are the elements of $\mathcal{T}$ and $T, T' \in \...
2
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0answers
123 views

Hamiltonian Matroids

Similar to graphs, a Matroid $M$ is said to be Hamiltonian if there is a base $B$ of $M$ and $e \in M-B$ such that $B + e$ is a cycle of $M$. Is there any literature on this? EDIT: Actually my ...
5
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1answer
250 views

Checking if a matroid is binary(Detecting $U^2_4$ minor in a matroid)

I am wondering what is the (computationally) best way to tell if a matroid of size $n$ and rank $r$ is binary(or whether it has a $U^2_4$ minor) given either one of these: 1) An independence oracle 2) ...
4
votes
1answer
511 views

Is there a graph-theoretical proof of Tutte's theorem on matroids?

First of all, I'm not a mathematician and I hope this question isn't too elementary, but I got no answers on math.SE, and since this is a reference request on a relatively advanced theorem, I thought ...
2
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0answers
169 views

Schemes defined by a collection of Plücker coordinates

If $C \subset {[n]\choose k}$ is any collection of $k$-element sets, we can define a scheme $$ W(C) = \bigcap_{S\notin C} \{V \in Gr(k,n) : p_S(V)=0\} \qquad \subseteq Gr(k,n), $$ where $p_S$ is the ...
3
votes
1answer
181 views

What is the name of this measure of matrix “degenerateness”

Given a spanning set, consider the minimum number of vectors that you must remove in order to make it no longer span. What is this number called? If the vectors are columns in a matrix $\Phi$, then ...
4
votes
0answers
344 views

Checking whether an element is in all inclusion-wise maximal common independent sets of two matroids

Given two matroids $M$ and $M'$ over the same universe $E$, and some element $x \in E$, I am interested in the importance of $x$ for the intersection (the common independent sets) of $M$ and $M'$. It ...
2
votes
1answer
138 views

Realizability of extensions of a free oriented matroid by an independent set

Question: I am searching for a non-realizable matroid with few dependencies relative to the number of points. Precisely, I would like to find a non-realizable (over $\mathbb{R}$) oriented matroid $M$ ...
2
votes
0answers
416 views

Incremental minimum spanning tree

Given a connected graph $G=(V,E)$ with a weight function $w:E\to\mathbb{R}$ and a subset $E_0\subseteq E$ such that the subgraph $(V,E_0)$ is connected, I am looking for a sequence $E_0\subseteq E_1\...
13
votes
1answer
889 views

Is there a Sudoku matroid?

This question is inspired from this one, where it is asked what is the minimum number of checks needed to verify that a Sudoku solution is correct. Let $$ E=\{r_1, \dots, r_9\} \cup \{c_1, \dots, ...
3
votes
2answers
284 views

are there pairs of combinatorial graphs that are both isospectral and have the same matroid?

Two graphs are isospectral if the combinatorial Laplacian on them has the same spectrum, equivalently, the adjacency matrix has the same the set of eigenvalues (including multiplicities). Two graphs ...
3
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0answers
168 views

A non-matroidal notion of dependence on a set of ideals

Assume we are given a set of ideals $I_1, \dots, I_s$ in a commutative polynomial ring. Let's define a subset indexed by $A\subseteq [s] = \{ 1,2,\dots, s\}$ as dependent if there exists an $a\in A$ ...
4
votes
1answer
248 views

Matroid representable over $\mathbb{R}$ but not over $\mathbb{Q}$?

Does there exist a matroid that is representable over $\mathbb{R}$ but not over $\mathbb{Q}$? In particular, can one give a positive answer using a nonrational polytope, i.e., a combinatorial ...
6
votes
2answers
375 views

Finding the matroids with a specified set of non-bases

I'm a grad student in algebraic geometry, and I've encountered a problem which requires me to produce an algorithm involving matroids. Since this isn't my area of expertise, I'm hoping someone knows ...
14
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1answer
506 views

Smooth bases of matroids

Motivated by algebraic geometry, I've come up with a purely combinatorial definition within the theory of matroids. The question is: is this concept known? If you like matroids but not algebraic ...
3
votes
1answer
340 views

from affine matroid to measures

Let $S$ be an arbitrary finite spanning subset of $\mathbb{R}^d$ of cardinality $N$. Let $W(S)$ be the formal $\mathbb{R}$-vector space generated by all $d$-dimensional simplices (i.e. bases of the ...
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1answer
188 views

When do the invariant factors of a direct sum of matrices correspond to those of its summands?

(Tried asking this on math stackexchange, but no takers so far.) I'm trying to prove something about matroids, which I have reduced to the following question: Suppose I have a matrix $M$ which is a ...
6
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0answers
327 views

Higher K-theory of Orlik-Solomon algebras (and possible generalizations?)

This topic of this question is a bit outside my comfort zone, and I should say that my end goal is to really understand how much "graph theory" is captured by contraction-deletion relations. It seems ...
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1answer
218 views

Is a non-disjoint union of connected matroids always connected?

This is perhaps an easy question, but... Let $M$ be a matroid on a ground set $E$, and let $A$ and $B$ be non-disjoint subsets of $E$ such that $M|A$ and $M|B$ are both connected. Is $M|(A\cup B)$ ...
11
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6answers
3k views

Good introductory text book on Matroid Theory?

I am looking for a good text book on Matroid theory. Ideally, one that might be better suited to engineers than pure mathematicians...but any book that is well written/organized would do. I have ...