Questions tagged [matroid-theory]
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.
198 questions
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Is there a countably infinite pre-closure with no circuits and no co-circuits?
For any $X$ call $f:2^X\to 2^X$ a pre-closure on $X$ when $\small\forall S,Q\subseteq X[S\subseteq Q\implies S\subseteq f(S)\subseteq f(Q)]$ while the complement of $T\subseteq X$ is $T^{\complement}=...
- 2,626
0
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0
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86
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Modular cuts over hyperplanes
Suppose I have a matroid $\mathcal{M}$ of rank $r$, and $k<r$ distinct hyperplanes $H_{1},\ldots,H_{k}$ of $\mathcal{M}$. Is the modular cut the $H_{i}$ generate proper?
- 53
0
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0
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132
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Hyperplanes which generate a non-proper modular cut
Suppose I have a set of hyperplanes $\{H_{i}\}$ of a (finite) matroid $\mathcal{M}$ the smallest modular cut over which is non-proper. It doesn’t seem to be true in general that there exists an ...
- 53
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67
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Vector matroids and projections of vector spaces (reference and literature request)
I am new in matroid theory, and currently I am working with so-called matroids of a vector space.
Definition. Let $V \subseteq \mathbb F^n$ be a vector space. A matroid of subspace $\mathbf M(V)$ is a ...
- 123
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0
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84
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Does the semigroup of covolume polynomials have the cancellation property?
Let $V_n^d(\mathbb{Q})$ denote the set of realizable volume polynomials of degree $d$ in $n$ variables over $\mathbb{Q}$ - these are polynomials of the form $\frac{1}{d!}\int_Y (\sum x_i D_i)^d$ where ...
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0
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124
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Is there an interesting connection between matroid theory and the Modular Group?
I'm presenting a seminar on the modular group to some friends, in a weekly seminar where we talk about any random mathematical thing we find interesting. Due to the ubiquity of appearances of the ...
- 447
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0
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95
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Is there a canonical form to test 2-isomorphism of graphs via cycle matroids?
Is there a canonical form that can be used to determine whether two graphs have isomorphic cycle matroids?
Input: Two unlabeled multigraphs $G_1$ and $G_2$.
Output: I want to check whether the cycle ...
- 101
3
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0
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82
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M-chain containing a given modular flat in a geometric lattice
Let $\mathcal{L}$ be a supersolvable geometric lattice and let $F$ be a modular element of $\mathcal{L}$. Does $\mathcal{L}$ necessarily admit a maximal chain of modular elements with $F$ in it ?
If $...
- 111
4
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2
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220
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Simpler proof of arborescence packing and isolating cut equivalence
Let $G$ be a directed graph with specified vertex $v$. We define a $v$-cut in $G$ to be a set of edges whose deletion from $G$ results in a graph where some vertex $w\neq v$ cannot be reached by a ...
- 599
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112
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Number of central $3$-dim hyperplane arrangements in generic position
Let $a(n)$ be the number of equivalence classes of $3$-dimensional central arrangements of $n$ hyperplanes in general position.
I believe that $a(n)= 1$ for $n \leq 5$ and $a(6)=3$. Is this correct? ...
- 1,912
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0
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80
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Balanced cocircuit cover
Are there studies on matroids which can be covered by $r$ cocircuits ($r$ is the rank of the matroid), so each element is covered by a small number of times?
For example, it is known graphic matroids ...
- 613
4
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0
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153
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Definition of Loop in an Oriented Matroid
I had posted this on Stackexchange because I don't believe this is a particlarly difficult question, but there were no answers, so I'm posting it on here now.
I just had a quick question about the ...
- 61
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Finite projective geometry and the Krasner hyperfield
The Krasner hyperfield is an algebraic structure of two operations on $K=\{0,1\}$ called $+\colon K\times K\to \mathcal{P}(K)$ and $\cdot\colon K\times K\to K$ with
$0+0=0$
$0+1=1+0=1$
$1+1=\{0,1\}$
...
- 11.1k
2
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1
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291
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Number of binary matroids of rank $r$ on a ground set with $n$ elements
How many simple binary matroids are there, up to isomorphism, of rank $r$ on an $n$-element ground set, where $r \le n < 2^r$? Write this number as $a_r(n)$. Is there somewhere where I can get this ...
- 49
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If the matroids associated to two finite subsets of the same vector space are isomorphic, are these two finite subsets linearly equivalent?
Let $E$ be a finite subset of ${\mathbb{F}_2}^n$, the $n$-dimensional vector space over the finite field $\mathbb{F}_2$ of $2$ elements. Let $M_E$ denote the associated matroid on $E$ where the ...
- 49
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0
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166
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All matroid polytope is a generalized permutohedron
In many texts, the authors say something like "a matroid polytope lives in the family of generalized permutohedra". We can quickly check the veracity of this claim by describing the matroid ...
- 261
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0
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133
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A generalized permutohedron as the sum of the dilatations of the faces of the standard simplex
I am trying to understand the proof of the statement, specifically it refers to a theorem stated by Postnikov in his text on permutohedra. So, this sentence claims the following:
If $\{Y_I \}$ is a ...
- 261
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1
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903
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Road map and references for combinatorial Hodge theory
I'm a PhD student. I'm familiar with graduate level algebraic geometry and toric varieties.
I wanted to know a road map for getting into combinatorial Hodge theory and other prerequisites that I'll ...
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115
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Is there an efficient algorithm for finding a fundamental cycle basis of a graph with the fewest odd cycles? Failing that, a hardness result on this?
I can think of a greedy algorithm:
Let $B$ be a fundamental cycle basis of graph $G$ induced by spanning tree (or forest) $T$
For $e\in T$, let $n_+(e)$ ($n_-(e)$) be the number of even (odd) cycles ...
- 33
4
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0
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158
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Find necessary & sufficient conditions for two families of sets to have $m$ pairwise disjoint common partial transversals of given sizes
Let $S$ be a finite set, $I$ a finite index set, $\mathcal A=(A_i:i\in I)$ and $\mathcal B=(B_i:i\in I)$ families of subsets of $S$.
For $J\subseteq I$, let $A(J)$ denote $\bigcup_{j\in J} A_j$.
A ...
- 1,979
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1
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Reconstructing a matroid by its minors
Proposition 3.1.27 in Oxley's Matroid Theory says that given a matroid $M$ and an element $e\in E(M)$ such that $e$ is not a loop or a coloop, the pair $(M/e, M\setminus e)$ uniquely determines $M$. ...
- 61
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1
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225
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The Salvetti complex of a non-realizable oriented matroid
Given a real hyperplane arrangement, the Salvetti complex of the associated oriented matroid is homotopy equivalent to the complement of the complexification of the arrangement. In particular, its ...
- 2,917
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2
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391
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Lattice description of matroid duality
Apologies for this very basic question in matroid theory, but I could not find anything about it online after a bit of searching.
There is a well-known bijective correspondence ("cryptomorphism&...
- 26k
6
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314
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A matroid parity exchange property
As part of my research, I encountered the following problem. Let $M = (E,I)$ be a matroid and let $P = \{P_1,\ldots,P_n\}$ be a partition of $E$ into (disjoint) pairs. For $A \subseteq P$, we say that ...
- 163
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2
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A generalized matroid exchange property
Let $(E,I)$ be a matroid, and let $A,B \in I$ be disjoint independent sets in the matroid. Moreover, let $B_1,\ldots, B_k$ be a partition of $B$. I could not decide if the following is always true. ...
- 163
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Book for matroid polytopes
I have made a study of polytopes with the books of Ziegler and "Integer Programming" of Conforti, my main goal is to study matroid polytopes; to study matroids I have thought about the book &...
- 261
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1
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Does a matroid base polytope contain its circumcenter?
Let $(X,\mathcal B)$ be a matroid on the ground set $X=(x_1,...,x_n)$ and with set of bases $\mathcal B$, and let $P\subset\Bbb R^n$ be its matroid base polytope (i.e. the convex hull of the ...
- 14.5k
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2
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430
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"Minimal" connected matroids
I'm interested in connected matroids $M$ on the ground set $[n]$ for which there is no connected matroid on $[n]$ of the same rank but with a strictly smaller set of bases (by inclusion). Equivalently,...
- 3,916
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Minimal matroid of rank r size n
We can define a partial order $\leq$ on loopless matroids, such that $M_1\leq M_2$ if $M_1$ and $M_2$ are on the same groundset and $B_1\subseteq B_2$, where $B_1$ and $B_2$ are the set of bases of $...
- 613
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How many linear matroids are transversal
It is known that almost all matroids are not linear matroids (a.k.a. not representable matroids). This was shown by Nelson:
arXiv: Almost all matroids are non-representable
A transversal matroid is a ...
- 7,933
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84
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Properties of the "girth function" of a matroid
Given an independent set representation of a matroid $M=(E,\mathcal{F})$ its ``rank function'' $r$ defined on the powerset of $E$ is:
$$
\forall X \subseteq E, \quad r(X) = \max_{Y \subseteq X}\{|Y|, ...
- 7,092
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0
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Adding columns to a binary matroid to make it graphic
It is well-known that graphic matroids are binary. Now suppose we have a binary non-graphic matroid $M$ with representing matrix $A$ over the field $\mathbb{Z}_2$. Is there a known way to add a small ...
- 7,092
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0
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97
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Are all real-representable matroids binary?
I doubt this is true but I was not able to find a clear answer to the question. Surely this is due to my erratic knowledge of matroid theory.
(I know the $U_4^2$-forbidden characterization, but I am ...
- 7,092
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107
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Homology of infinite matroids of finite rank
Bjorner has a great paper about the homology of independence complexes of finite matroids, which is the usual context in matroid theory as far as I understand. However, I've also been told that often ...
- 2,251
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2
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381
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Is the 3-sum of two graphic matroids a graphic matroid?
A regular matroid is a matroid which is representable over any field. It is a famous theorem of Seymour's that the any regular matroid is obtained by performing 1,2, and 3 sums on graphic, cographic ...
- 155
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How to prove the local search algorithm can find the maximum weight independent set in a matroid with cardinality constraint?
I am trying to prove a simple local search algorithm could solve exactly this problem:
$\underset{S \in I(M), |S|=k}{max} c(S)$
where $M$ is a matroid, and $ I(M)$ is the set of all independent set, $...
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Minimum number of matroid circuits containing a fixed element
Let $M$ be a matroid with an $n$-element ground set $E$. I'll assume that $M$ is connected, co-simple (so its dual has no loops or parallel elements) and has no loops. Fix a particular element $e\in E$...
- 145
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Algorithm for finding a minimum weight circuit in a weighted binary matroid
For a given weighted graph $G = (V, E)$, there is a simple algorithm for finding the minimum weight circuit by running Dijkstra's algorithm $|E|$ times.
Also for a matroid $M = (E, I)$ one can use the ...
- 33
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258
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A Plücker coordinate matroid
Let $V$ be an $n$-dimensional vector space over a field $F$. Let
$\mathrm{Gr}(V,d)$ be the set (Grassmann variety) of all
$d$-dimensional subspaces of $V$. We can regard
$\mathrm{Gr}(V,d)$ as a subset ...
- 53.7k
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107
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Topology of independence set of a vector space
This seems like something that would have a well-known treatment somewhere, but I'm not sure where to look. If we have a vector space $V$ (or maybe even a module), we can consider an abstract ...
- 2,251
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76
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Associating a matroid to a uniform hypergraph
For a fixed ground set $[n]=\{1,\ldots,n\}$, and for any matroid $M$ on $[n]$, specified as a collection of bases $B_M$, the corresponding matroid basis polytope $P_M$ is defined to be the convex hull ...
3
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1
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399
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When Alexander dual of a simplicial complex is a matroid?
Let $C$ be a simplicial complex on a finite set $V$: that means $C$ is a collection of subsets of $V$ such that if $\sigma\in C$ and $\tau\subseteq \sigma$, then $\tau\in C$.
The Alexander dual $D(C)$ ...
- 551
1
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1
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Connectivity of a matroid is at least its rank?
The connectivity $\eta(X)$ of a simplicial complex $X$ is defined as the
$$1+\min_j\{j \mid \tilde{H}_j(X)\neq 0\}.$$
If no such $j$ exists, then $\eta(X):=\infty$.
(See here for this definition, ...
- 551
6
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1
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406
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Distributive lattice of subspaces
Let $V$ be a finite dimensional vector space. Let $\Lambda$ be a collection of subspaces of $V$ such that, if $X$ and $Y$ are in $\Lambda$, then $X\cap Y$ and $X+Y$ are in $\Lambda$. This makes $\...
- 161k
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213
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Rowmotion of matroids
If $Z$ is a finite poset, then we say that a collection $\mathcal{A}$ is an antichain if whenever $y,z\in\mathcal{A}$, if $y\leq z$, then $y=z$. If $R\subseteq Z$, then let $L(R)$ be the set of all $x\...
- 29.1k
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5
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1k
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Birkhoff's representation theorem vs matroid-geometric lattice correspondence
This question is motivated by the superficial observation that Birkhoff's representation theorem and the cryptomorphism between matroids and geometric lattices are sort of similar. The former says ...
- 3,916
3
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0
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149
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Matroid of monomials of linear forms
Consider the linear matroid $M(k, d)$ on the monomials of degree $d$ in $k$ general linear forms $L_1, \ldots, L_k$ in two variables, over $\mathbb{C}$. For simplicity take $k=3$ and the forms $X, Y, ...
- 740
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0
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133
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$r(M)$-subsets of a 3-connected matroid $M$
It is proved in Lowrance, Oxley, Semple, and Welsh - On properties of almost all matroids that almost all matroids are 3-connected asymptotically. Also, it is conjectured that almost all matroids are ...
- 379
3
votes
1
answer
207
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Nontrivial upper bounds for the nullity of hyperplanes in paving matroids
$\DeclareMathOperator\null{null}$Let $H$ be a hyperplane of the paving matroid $M$ with $r(M)=n$. How large can $\null(H)$ be?
We know that $\null(H)=|H|-r(H)=|H|-(n-1)$. So everything boils down to ...
- 379
2
votes
1
answer
152
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Counting certain bases of a paving matroids
Let $M=(E,I)$ be a paving matroid with rank $n$. Let $A\subset E$ be an $n-1$ subset. How many bases of $M$ containing $A$ exist? (Note that every $n-1$ subset of $E$ is independent.)
- 379