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On characterizations of binary and graphic matroids

Jean-Paul Doignon
1981 Discrete Mathematics  
Short proofs are presented for two results due respectively to Tutte and Welsh.  ...  We now turn to Welsh's characterization of graphic matroids among binary ones. (Welsh [l]). A binary matroid M on a set S is graphic ifl it admits a 2-complete basis (Ci)iEl of coci&ts. . Suficiency.  ...  We give here a short proof of this rest;;, and also a direct argument for a characterization of graphic matroids stated by Welsh (see below).  ... 
doi:10.1016/0012-365x(81)90229-6 fatcat:dymoxaqbzrfsngkkgykj65ujxm

On graphic splitting of regular matroids [article]

Ganesh Mundhe, K. V. Dalvi
2020 arXiv   pre-print
We study the effect of the splitting operation on regular matroids and characterize the class of regular matroids which yield graphic matroids under the splitting operation.  ...  Raghunathan at al. [9] introduced splitting operation with respect to a pair of element for binary matroid and characterized Eulerian binary matroids using it.  ...  Suppose M contains a minor isomorphic to one of the matroids M(G 1 ), M(G 2 ) and M(G 3 ). Then, by Theorem 2.7, M x,y is not cographic for some pair x, y of elements of M.  ... 
arXiv:2002.04214v2 fatcat:75wem4y2hjapjjc36ixhqgwwru

Base exchange properties of graphic matroids

Marcel Wild
1996 Discrete Mathematics  
New base exchange properties of binary and graphic matroids are derived. The graphic matroids within the class of 4-connected binary matroids are characterized by base exchange properties.  ...  Some progress with the characterization of arbitrary graphic matroids is made.  ...  Characterizing graphic matroids among 4-connected binary matroids by base exchange properties Let M be a binary matroid on the set E.  ... 
doi:10.1016/0012-365x(94)00171-e fatcat:4nkrz72s5fbe3boitns4v3uucq

Forbidden-minor characterization for the class of graphic element splitting matroids

Y.M. Borse, Kiran Dalvi, M.M. Shikare
2009 Discussiones Mathematicae Graph Theory  
In this paper, we prove that an element splitting operation by every pair of elements on a cographic matroid yields a cographic matroid if and only if it has no minor isomorphic to M (K 4 ).  ...  Dalvi, Borse and Shikare [3] characterized graphic matroids whose element splitting matroids are also graphic as follows. Theorem 1.1.  ...  The element splitting operation, by any pair of elements, on a graphic matroid yields a graphic matroid if and only if it has no minor isomorphic to M (K 4 ), where K 4 is the complete graph on 4 vertices  ... 
doi:10.7151/dmgt.1469 fatcat:5rdgc4yxnvcl5kib3rb4yuccoi

Forbidden-minor characterization for the class of cographic element splitting matroids

Y.M. Borse, Kiran Dalvi, M.M. Shikare
2011 Discussiones Mathematicae Graph Theory  
In this paper, we prove that an element splitting operation by every pair of elements on a cographic matroid yields a cographic matroid if and only if it has no minor isomorphic to M (K 4 ).  ...  Dalvi, Borse and Shikare [3] characterized graphic matroids whose element splitting matroids are also graphic as follows. Theorem 1.1.  ...  The element splitting operation, by any pair of elements, on a graphic matroid yields a graphic matroid if and only if it has no minor isomorphic to M (K 4 ), where K 4 is the complete graph on 4 vertices  ... 
doi:10.7151/dmgt.1568 fatcat:yhodppijcbdlnhpo5lwsie6i6e

Combinatorial geometries

1989 Discrete Mathematics  
matroids. 2.6 Special classes of binary matroids; graphic matroids. 2.7 Appendix on modular pairs of circuits in a matroid. 3.  ...  Fournier. 2.1 Definition and basic properties. 2.2 Characterizations of binary matroids. 2.3 Related characterizations. 2.4 Spaces of circuits of binary matroids. 2.5 Coordinatixing matrices of binary  ... 
doi:10.1016/0012-365x(89)90315-4 fatcat:bwi77zqc55gwbdog2eaotnods4

An introduction to coding sequences of graphs [article]

Shamik Ghosh, Raibatak Sen Gupta, M. K. Sen
2016 arXiv   pre-print
In his pioneering paper on matroids in 1935, Whitney obtained a characterization for binary matroids and left a comment at end of the paper that the problem of characterizing graphic matroids is the same  ...  Moreover considering coding sequences as binary matroids, we obtain a characterization for simple graphic matroids and found a necessary and sufficient condition for graph isomorphism in terms of a special  ...  In his pioneering paper on matroids in 1935, Whitney obtained a characterization for binary matroids and left a comment at end of the paper that the problem of characterizing graphic matroids is the same  ... 
arXiv:1505.04602v4 fatcat:fqih7aeasjao3awis2zhijvqti

Graphic and cographic Γ-extensions of binary matroids

Y.M. Borse, Ganesh Mundhe
2018 Discussiones Mathematicae Graph Theory  
Slater introduced the point-addition operation on graphs to characterize 4-connected graphs. The Γ-extension operation on binary matroids is a generalization of the point-addition operation.  ...  In this paper, we obtain forbidden minor characterizations for binary matroids whose Γ-extension matroids are graphic (respectively, cographic). Definition 1 [1].  ...  Acknowledgments The authors are grateful to the referees for their fruitful suggestions to improve the quality of the paper. We thank the referee who suggested an alternative proof of Theorem 5.  ... 
doi:10.7151/dmgt.2043 fatcat:24urpnh5lzhvbl2al5r3zsw2mi

Chordal characterization of graphic matroids

Wiktor Piotrowski
1988 Discrete Mathematics  
One of the deepest theorems in the theory of matroids is Tuttes excludedminor characterization of graphic matroids [ 111.  ...  The purpose of this paper is to present the characterization of graphic matroids using the concept of a chord.  ...  A binary matroid M is graphic if and only if it does not contain any Necessity. Let M be a graphic matroid. There is a graph G such that = M(G).  ... 
doi:10.1016/0012-365x(88)90119-7 fatcat:tcbz77y7qvfznitakarnefxli4

On graphic elementary lifts of graphic matroids [article]

Ganesh Mundhe, Y. M. Borse, K. V. Dalvi
2019 arXiv   pre-print
Zaslavsky introduced the concept of lifted-graphic matroid. For binary matroids, a binary elementary lift can be defined in terms of the splitting operation.  ...  In this paper, we give a method to get a forbidden-minor characterization for the class of graphic matroids whose all lifted-graphic matroids are also graphic using the splitting operation.  ...  They defined the splitting operation for binary matroids with respect to a pair of elements and used it to characterize the binary Eulerian matroids. Later on, Shikare et al.  ... 
arXiv:1910.05689v1 fatcat:hohz2yhpizh5le5qw5kiidmy4q

Rainbow and monochromatic circuits and cuts in binary matroids [article]

Kristóf Bérczi, Tamás Schwarcz
2021 arXiv   pre-print
As a byproduct, we give a characterization of binary matroids in terms of reductions to partition matroids.  ...  Motivated by a conjecture of Bérczi et al., we also analyze the relation between the covering number of a binary matroid and the maximum number of colors or the maximum size of a color class in any of  ...  The proof is based on the excluded-minor characterization of binary matroids, so we first prove two lemmas about standard colorings of minors of matroids. Lemma 14.  ... 
arXiv:2012.05037v3 fatcat:p27yjvvz5jhv5kas72lb7pommq

Binary Supersolvable Matroids and Modular Constructions

Gunter M. Ziegler
1991 Proceedings of the American Mathematical Society  
Then we characterize the case that the modular join of two matroids is supersolvable. This is used to study modular flats and modular joins of binary supersolvable matroids.  ...  Let Jt be the class of binary matroids without a Fano plane as a submatroid. We show that every supersolvable matroid in JÍ is graphic, corresponding to a chordal graph.  ...  Acknowledgment The author is grateful to Bob Bixby, Dan Kleitman, and Klaus Truemper for valuable discussions. He thanks Doris Konnerth for moral and secretarial support.  ... 
doi:10.2307/2048620 fatcat:asfxmapwjfdezntz7qhm3wujxi

Binary supersolvable matroids and modular constructions

G{ünter M. Ziegler
1991 Proceedings of the American Mathematical Society  
Then we characterize the case that the modular join of two matroids is supersolvable. This is used to study modular flats and modular joins of binary supersolvable matroids.  ...  Let Jt be the class of binary matroids without a Fano plane as a submatroid. We show that every supersolvable matroid in JÍ is graphic, corresponding to a chordal graph.  ...  Acknowledgment The author is grateful to Bob Bixby, Dan Kleitman, and Klaus Truemper for valuable discussions. He thanks Doris Konnerth for moral and secretarial support.  ... 
doi:10.1090/s0002-9939-1991-1068134-3 fatcat:d6etfxmvn5cafkurahiv75hbnu

Some Results Involving the Splitting Operation on Binary Matroids

Naiyer Pirouz, Maruti Mukinda Shikare
2012 ISRN Discrete Mathematics  
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.  ...  Shikare and Waphare 7 characterized graphic matroids whose splitting matroids are also graphic. In fact, they proved the following theorem.  ...  Let M E, C be a binary matroid on a set E together with the set C of circuits and let {x, y} ⊆ E.  ... 
doi:10.5402/2012/406147 fatcat:t47nffw4bjd2xaclgwo7vvtlzm

On Mighton's characterization of graphic matroids

Donald K. Wagner
2010 Journal of combinatorial theory. Series B (Print)  
Both the present proof and Mighton's proof rely on Tutte's [8] recursive characterization of graphic matroids, which is based on structural properties of cocircuits and their bridges.  ...  Introduction Mighton [5] recently gave a very nice characterization of those binary matroids that are graphic. The purpose of this note is to provide a shorter proof of the result.  ... 
doi:10.1016/j.jctb.2010.01.005 fatcat:6ijsjq4l4jd2hlfqlkq6ue3vqm
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