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On characterizations of binary and graphic matroids
1981
Discrete Mathematics
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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]
2020
arXiv
pre-print
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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
1996
Discrete Mathematics
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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
2009
Discussiones Mathematicae Graph Theory
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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
2011
Discussiones Mathematicae Graph Theory
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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
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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]
2016
arXiv
pre-print
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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
2018
Discussiones Mathematicae Graph Theory
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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
1988
Discrete Mathematics
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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]
2019
arXiv
pre-print
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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]
2021
arXiv
pre-print
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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
1991
Proceedings of the American Mathematical Society
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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
1991
Proceedings of the American Mathematical Society
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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
2012
ISRN Discrete Mathematics
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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
2010
Journal of combinatorial theory. Series B (Print)
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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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