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.

On 3-connected es-splitting binary matroids

2016 ◽  
Vol 09 (01) ◽  
pp. 1650017 ◽  
Author(s):  
S. B. Dhotre ◽  
P. P. Malavadkar ◽  
M. M. Shikare
Keyword(s):  
Sufficient Condition ◽  
Binary Matroid ◽  
Binary Matroids ◽  
Splitting Operation

The es-splitting operation on a [Formula: see text]-connected binary matroid may not preserve the [Formula: see text]-connectedness of the matroid. In this paper, we provide a sufficient condition for a [Formula: see text]-connected binary matroid to yield a [Formula: see text]-connected binary matroid under the es-splitting operation. We derive a splitting lemma for [Formula: see text]-connected binary matroids as an application of this result.

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 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 Idealness of k-wise intersecting families

2020 ◽  
Author(s):  
Ahmad Abdi ◽  
Gérard Cornuéjols ◽  
Tony Huynh ◽  
Dabeen Lee
Keyword(s):  
Chromatic Number ◽  
Common Element ◽  
Binary Matroids ◽  
Projective Geometries ◽  
Intersecting Families ◽  
Cycle Covers ◽  
Binary Clutters

Abstract A clutter is k-wise intersecting if every k members have a common element, yet no element belongs to all members. We conjecture that, for some integer $$k\ge 4$$ k ≥ 4 , every k-wise intersecting clutter is non-ideal. As evidence for our conjecture, we prove it for $$k=4$$ k = 4 for the class of binary clutters. Two key ingredients for our proof are Jaeger’s 8-flow theorem for graphs, and Seymour’s characterization of the binary matroids with the sums of circuits property. As further evidence for our conjecture, we also note that it follows from an unpublished conjecture of Seymour from 1975. We also discuss connections to the chromatic number of a clutter, projective geometries over the two-element field, uniform cycle covers in graphs, and quarter-integral packings of value two in ideal clutters.

scholarly journals Some Results Involving the Splitting Operation on Binary Matroids

2012 ◽  
Vol 2012 ◽  
pp. 1-8
Author(s):  
Naiyer Pirouz ◽  
Maruti Mukinda Shikare
Keyword(s):  
Binary Matroids ◽  
Splitting Operation

We obtain some results concerning the planarity and graphicness of the splitting matroids. Further, we explore the effect of splitting operation on the sum of two matroids.

scholarly journals Circuit-Difference Matroids

10.37236/9314 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
George Drummond ◽  
Tara Fife ◽  
Kevin Grace ◽  
James Oxley
Keyword(s):  
Disjoint Union ◽  
Symmetric Difference ◽  
Binary Matroids ◽  
Regular Matroid

One characterization of binary matroids is that the symmetric difference of every pair of intersecting circuits is a disjoint union of circuits. This paper considers circuit-difference matroids, that is, those matroids in which the symmetric difference of every pair of intersecting circuits is a single circuit. Our main result shows that a connected regular matroid is circuit-difference if and only if it contains no pair of skew circuits. Using a result of Pfeil, this enables us to explicitly determine all regular circuit-difference matroids. The class of circuit-difference matroids is not closed under minors, but it is closed under series minors. We characterize the infinitely many excluded series minors for the class.

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