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Matching Problems with Delta-Matroid Constraints

Naonori Kakimura, Mizuyo Takamatsu
2014 SIAM Journal on Discrete Mathematics  
This paper introduces a class of the delta-matroid matching problem, where the given delta-matroid is a projection of a linear delta-matroid.  ...  Abstract Given an undirected graph G = (V, E) and a delta-matroid (V, F), the delta-matroid matching problem is to find a maximum cardinality matching M such that the set of the end vertices of M belongs  ...  Acknowledgement The authors are grateful to Satoru Iwata for fruitful comments on this topic, and Maiko Shigeno for informing us of the recent paper [1] .  ... 
doi:10.1137/110860070 fatcat:llpjo74chrfltgiqcuobs2eiei

The linear delta-matroid parity problem

James F. Geelen, Satoru Iwata, Kazuo Murota
2003 Journal of combinatorial theory. Series B (Print)  
This paper addresses a generalization of the matroid parity problem to delta-matroids. We give a minimax relation, as well as an efficient algorithm, for linearly represented deltamatroids.  ...  These are natural extensions of the minimax theorem of Lova´sz and the augmenting path algorithm of Gabow and Stallmann for the linear matroid parity problem. r  ...  Acknowledgments We are grateful to Andre´Bouchet and Andra´s Seb + o for helpful comments, which resulted in a substantial improvement of the presentation in Section 5.  ... 
doi:10.1016/s0095-8956(03)00039-x fatcat:r7fmsok5hzbydmii6wsm26po3m

Matroid Matching Via Mixed Skew-Symmetric Matrices

James Geelen, Satoru Iwata
2005 Combinatorica  
This modest generalization of the matching problem contains the linear matroid matching problem and, more generally, the linear delta-matroid parity problem.  ...  This matrix, called the Tutte matrix, has rank exactly twice the size of a maximum cardinality matching of G. Thus, to find the size of a maximum matching it suffices to compute the rank of T .  ...  Similar techniques prove the following extension of Tutte's theorem. 1991 Mathematics Subject Classification 05C70. Keywords: matching, matroids, delta-matroids, matroid parity  ... 
doi:10.1007/s00493-005-0013-7 fatcat:hfrodg5apvfhbgpw5wq2depw4m

Page 53 of Mathematical Reviews Vol. , Issue 2001A [page]

2001 Mathematical Reviews  
In particular, two delta-matroids defined on the same set give rise to a parity sys- tem; the essential idea is to add the rank functions of the given delta-matroids.  ...  a delta-matroid.  ... 

Parity Systems and the Delta-Matroid Intersection Problem

André Bouchet, Bill Jackson
1999 Electronic Journal of Combinatorics  
We consider the problem of determining when two delta-matroids on the same ground-set have a common base.  ...  Examples of parity systems may be obtained by combining either, two delta-matroids, or two orthogonal 2-polymatroids, on the same ground-sets.  ...  This contradicts the choice of M and completes the proof of the proposition. Sums of two 2-matroids Let Q 1 and Q 2 be 2-matroids.  ... 
doi:10.37236/1492 fatcat:nyyqyrpmsffhrcyz4yagal5axm

Delta-Matroids, Jump Systems, and Bisubmodular Polyhedra

André Bouchet, William H. Cunningham
1995 SIAM Journal on Discrete Mathematics  
We also prove addition and composition theorems for jump systems, which have several applications for delta-matroids and matroids.  ...  Unlike the case for usual submodularity, the points of interest are not all the integral points in the relevant polyhedron, but form a subset of them.  ...  Matching delta-matroids are examples of linkable delta-matroids. But for matching delta-matroids an even stronger property holds.  ... 
doi:10.1137/s0895480191222926 fatcat:lxsspqnwbffrbdvoy3w6kmt7bq

Page 6109 of Mathematical Reviews Vol. , Issue 98J [page]

1998 Mathematical Reviews  
Volkmar Welker (Berlin) 98j:05044 05B35 Bouchet, André (F-LMNS-I; Le Mans); Schwarzler, Werner (D-BONN-DM; Bonn) The delta-sum of matching delta-matroids.  ...  A delta-matroid is tractable if an algorithm is known to evaluate the separation oracle in polynomial time. The delta-sum of the delta-matroids (V,¥,) and (V,¥>) is the delta-  ... 

An extension of Stanley's chromatic symmetric function to binary delta-matroids [article]

Marina Dudina, Vyacheslav Zhukov
2018 arXiv   pre-print
polynomial of binary delta-matroids we define satisfies the 4-term relations for binary delta-matroids [10] and determines, therefore, an invariant of links.  ...  Instead, we consider the binary delta-matroid associated to an embedded graph and define the extended Stanley's chromatic polynomial as an invariant of delta-matroids.  ...  It suffices to take into account that the value of the character ξ B on an even binary delta-matroid is non-zero only if this delta-matroid is the product of one-element binary delta-matroids.  ... 
arXiv:1809.03431v1 fatcat:bz3qx6iqp5el3jsjlqcdp24rbu

Delta matroids whose fundamental graphs are bipartite

Alain Duchamp
1992 Linear Algebra and its Applications  
We consider a generalization of finite matroids called delta matroids. This structure has been introduced simultaneously by A. Bouchet and, with pseudoma-  ...  (ii) Env(G') is the direct sum of two delta matroid whose height is 1. set. X is a cocircuit if X-is a circuit.  ...  We suppose 1 = 2, and we prove that Env(G) is a direct sum of two delta matroids whose height is 1.  ... 
doi:10.1016/0024-3795(92)90441-c fatcat:7d6hlvix7fdy5iktfj3vynqfc4

The Interlace Polynomial [article]

Ada Morse
2016 arXiv   pre-print
systems and delta-matroids.  ...  In this paper, we survey results regarding the interlace polynomial of a graph, connections to such graph polynomials as the Martin and Tutte polynomials, and generalizations to the realms of isotropic  ...  The interlace polynomial of a delta-matroid also satisfies a recurrence generaliz- ing that of the interlace polynomial of a graph. Theorem 22. [22] Let D = (E, F) be a delta-matroid.  ... 
arXiv:1601.03003v1 fatcat:eqkt72dg3fgj5hmcxdh2dh2gti

Nullity and Loop Complementation for Delta-Matroids

Robert Brijder, Hendrik Jan Hoogeboom
2013 SIAM Journal on Discrete Mathematics  
., the delta-matroids obtained after loop complementation and after pivot on a single element together with the original delta-matroid fulfill the property that two of them have equal "null space" while  ...  ., symmetric matrices over GF(2)) may be seen as a special class of delta-matroids, this distance measure generalizes the notion of nullity in this case.  ...  We thank Gordon Royle for kindly sending to us the database of matroids with nine elements from [19] .  ... 
doi:10.1137/110854692 fatcat:aoyrayvicbc7pbzl3grtuwocey

Canonical binary Δ-matroids [article]

Rémi Cocou Avohou, Brigitte Servatius, Herman Servatius
2022 arXiv   pre-print
We provide a canonical form for binary delta-matroids without restriction on the feasibility of the empty set.  ...  Mphako-Bandab, who show that, using a delta-matroid analogue of handle slides, every binary delta-matroid in which the empty set is feasible can be written in a canonical form analogous to the canonical  ...  Therefore, D is isomorphic to the ∆-matroid obtained by taking the direct sum of the ∆-matroid D /F ⊕(F, {F }).  ... 
arXiv:2203.04365v1 fatcat:b4y2frtq5ne6vhhitgs2z3uw3y

Isotropic Matroids I: Multimatroids and Neighborhoods

Robert Brijder, Lorenzo Traldi
2016 Electronic Journal of Combinatorics  
Several properties of the isotropic matroid of a looped simple graph are presented.  ...  Results include a characterization of the multimatroids that are associated with isotropic matroids and several ways in which the isotropic matroid of $G$ incorporates information about graphs locally  ...  Acknowledgements We thank the referee for careful readings of preliminary versions of the paper, and many useful comments.  ... 
doi:10.37236/5222 fatcat:htzpxbbxxrbv7j3av5prex5sge

Author index to volume 181 (1998)

1998 Discrete Mathematics  
Schw~irzler, The delta-sum of matching delta-matroids (1-3 Brandt, S., see D. Amar , M. and A.V. Karzanov, Minimum weight (T, d)-joins and multi-joins (1-3 Burzio, M. and G.  ...  Schoen, Independent finite sums in graphs defined on the natural numbers (Note) , A.N., On k-weak orders: Recognition and a tolerance result (1-3) Tsai, Y.S., see H.J. Shyr (1-3)  ... 
doi:10.1016/s0012-365x(97)82062-6 fatcat:sqdp43rjovam5az5gxnwxknhxq

Isotropic matroids I: Multimatroids and neighborhoods [article]

Robert Brijder, Lorenzo Traldi
2016 arXiv   pre-print
Several properties of the isotropic matroid of a looped simple graph are presented.  ...  Results include a characterization of the multimatroids that are associated with isotropic matroids and several ways in which the isotropic matroid of G incorporates information about graphs locally equivalent  ...  Acknowledgements We thank the referee for careful readings of preliminary versions of the paper, and many useful comments.  ... 
arXiv:1503.04406v5 fatcat:woqe5h2rpbdlniz2sjqwwdc4ka
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