scholarly journals Minors of a random binary matroid

2019 ◽  
Vol 55 (4) ◽  
pp. 865-880
Author(s):  
Colin Cooper ◽  
Alan Frieze ◽  
Wesley Pegden
Keyword(s):  
Binary Matroid

scholarly journals On Binary Matroid Minors and Applications to Data Storage over Small Fields

2017 ◽  
pp. 139-153
Author(s):  
Matthias Grezet ◽  
Ragnar Freij-Hollanti ◽  
Thomas Westerbäck ◽  
Camilla Hollanti
Keyword(s):  
Data Storage ◽  
Binary Matroid ◽  
Small Fields

A characterization of n-connected splitting matroids

2014 ◽  
Vol 07 (04) ◽  
pp. 1450060
Author(s):  
P. P. Malavadkar ◽  
M. M. Shikare ◽  
S. B. Dhotre
Keyword(s):  
Binary Matroid ◽  
Binary Matroids ◽  
Splitting Operation

The splitting operation on an n-connected binary matroid may not yield an n-connected binary matroid. In this paper, we characterize n-connected binary matroids which yield n-connected binary matroids by the generalized splitting operation.

scholarly journals Splitting in a binary matroid

1998 ◽  
Vol 184 (1-3) ◽  
pp. 267-271 ◽  
Author(s):  
T.T. Raghunathan ◽  
M.M. Shikare ◽  
B.N. Waphare
Keyword(s):  
Binary Matroid

scholarly journals Quasiregular Matroids

10.37236/6911 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
S. R. Kingan
Keyword(s):  
The Self ◽  
Binary Matroid ◽  
Binary Matroids

Regular matroids are binary matroids with no minors isomorphic to the Fano matroid $F_7$ or its dual $F_7^*$. Seymour proved that 3-connected regular matroids are either graphs, cographs, or $R_{10}$, or else can be decomposed along a non-minimal exact 3-separation induced by $R_{12}$. Quasiregular matroids are binary matroids with no minor isomorphic to the self-dual binary matroid $E_4$. The class of quasiregular matroids properly contains the class of regular matroids. We prove that 3-connected quasiregular matroids are either graphs, cographs, or deletion-minors of $PG(3,2)$, $R_{17}$ or $M_{12}$ or else can be decomposed along a non-minimal exact 3-separation induced by $R_{12}$, $P_9$, or $P_9^*$.

scholarly journals Playing Nim on a Simplicial Complex

10.37236/1233 ◽  
1996 ◽  
Vol 3 (1) ◽  
Author(s):  
Richard Ehrenborg ◽  
Einar Steingrímsson
Keyword(s):  
Simplicial Complex ◽  
Winning Strategy ◽  
Independent Sets ◽  
Binary Matroid ◽  
Classical Game

We introduce a generalization of the classical game of Nim by placing the piles on the vertices of a simplicial complex and allowing a move to affect the piles on any set of vertices that forms a face of the complex. Under certain conditions on the complex we present a winning strategy. These conditions are satisfied, for instance, when the simplicial complex consists of the independent sets of a binary matroid. Moreover, we study four operations on a simplicial complex under which games on the complex behave nicely. We also consider particular complexes that correspond to natural generalizations of classical Nim.

scholarly journals A Characterization of Circle Graphs in Terms of Multimatroid Representations

10.37236/6992 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Robert Brijder ◽  
Lorenzo Traldi
Keyword(s):  
Simple Graph ◽  
Rank Function ◽  
Binary Matroid ◽  
Circle Graph ◽  
Circle Graphs ◽  
Isotropic System

The isotropic matroid $M[IAS(G)]$ of a looped simple graph $G$ is a binary matroid equivalent to the isotropic system of $G$. In general, $M[IAS(G)]$ is not regular, so it cannot be represented over fields of characteristic $\neq 2$. The ground set of $M[IAS(G)]$ is denoted $W(G)$; it is partitioned into 3-element subsets corresponding to the vertices of $G$. When the rank function of $M[IAS(G)]$ is restricted to subtransversals of this partition, the resulting structure is a multimatroid denoted $\mathcal{Z}_{3}(G)$. In this paper we prove that $G$ is a circle graph if and only if for every field $\mathbb{F}$, there is an $\mathbb{F}$-representable matroid with ground set $W(G)$, which defines $\mathcal{Z}_{3}(G)$ by restriction. We connect this characterization with several other circle graph characterizations that have appeared in the literature.

On Vertex-Triads in 3-Connected Binary Matroids

1998 ◽  
Vol 7 (4) ◽  
pp. 485-497 ◽  
Author(s):  
HAIDONG WU
Keyword(s):  
Connected Graph ◽  
Binary Matroid ◽  
Binary Matroids

A cocircuit C* in a matroid M is said to be non-separating if and only if M[setmn ]C*, the deletion of C* from M, is connected. A vertex-triad in a matroid is a three-element non-separating cocircuit. Non-separating cocircuits in binary matroids correspond to vertices in graphs. Let C be a circuit of a 3-connected binary matroid M such that [mid ]E(M)[mid ][ges ]4 and, for all elements x of C, the deletion of x from M is not 3-connected. We prove that C meets at least two vertex-triads of M. This gives direct binary matroid generalizations of certain graph results of Halin, Lemos, and Mader. For binary matroids, it also generalizes a result of Oxley. We also prove that a minimally 3-connected binary matroid M which has at least four elements has at least ½r*(M)+1 vertex-triads, where r*(M) is the corank of the matroid M. An immediate consequence of this result is the following result of Halin: a minimally 3-connected graph with n vertices has at least 2n+6/5 vertices of degree three. We also generalize Tutte's Triangle Lemma for general matroids.

scholarly journals Isotropic Matroids III: Connectivity

10.37236/5937 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Robert Brijder ◽  
Lorenzo Traldi
Keyword(s):  
Pendant Vertex ◽  
Split Decomposition ◽  
Binary Matroid ◽  
Isotropic System ◽  
Interesting Theorem

The isotropic matroid $M[IAS(G)]$ of a graph $G$ is a binary matroid, which is equivalent to the isotropic system introduced by Bouchet. In this paper we discuss four notions of connectivity related to isotropic matroids and isotropic systems. We show that the isotropic system connectivity defined by Bouchet is equivalent to vertical connectivity of $M[IAS(G)]$, and if $G$ has at least four vertices, then $M[IAS(G)]$ is vertically 5-connected if and only if $G$ is prime (in the sense of Cunningham's split decomposition). We also show that $M[IAS(G)]$ is $3$-connected if and only if $G$ is connected and has neither a pendant vertex nor a pair of twin vertices. Our most interesting theorem is that if $G$ has $n\geq7$ vertices then $M[IAS(G)]$ is not vertically $n$-connected. This abstract-seeming result is equivalent to the more concrete assertion that $G$ is locally equivalent to a graph with a vertex of degree $<\frac{n-1}{2}$.

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