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Matching Problems with Delta-Matroid Constraints

2014
*
SIAM Journal on Discrete Mathematics
*

This paper introduces a class

doi:10.1137/110860070
fatcat:llpjo74chrfltgiqcuobs2eiei
*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] . ...##
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The linear delta-matroid parity problem

2003
*
Journal of combinatorial theory. Series B (Print)
*

This paper addresses a generalization

doi:10.1016/s0095-8956(03)00039-x
fatcat:r7fmsok5hzbydmii6wsm26po3m
*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. ...##
###
Matroid Matching Via Mixed Skew-Symmetric Matrices

2005
*
Combinatorica
*

This modest generalization

doi:10.1007/s00493-005-0013-7
fatcat:hfrodg5apvfhbgpw5wq2depw4m
*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 ...##
###
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*. ...##
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Parity Systems and the Delta-Matroid Intersection Problem

1999
*
Electronic Journal of Combinatorics
*

We consider

doi:10.37236/1492
fatcat:nyyqyrpmsffhrcyz4yagal5axm
*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*. ...##
###
Delta-Matroids, Jump Systems, and Bisubmodular Polyhedra

1995
*
SIAM Journal on Discrete Mathematics
*

We also prove addition and composition theorems for jump systems, which have several applications for

doi:10.1137/s0895480191222926
fatcat:lxsspqnwbffrbdvoy3w6kmt7bq
*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. ...##
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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*- ...##
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An extension of Stanley's chromatic symmetric function to binary delta-matroids
[article]

2018
*
arXiv
*
pre-print

polynomial

arXiv:1809.03431v1
fatcat:bz3qx6iqp5el3jsjlqcdp24rbu
*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*. ...##
###
Delta matroids whose fundamental graphs are bipartite

1992
*
Linear Algebra and its Applications
*

We consider a generalization

doi:10.1016/0024-3795(92)90441-c
fatcat:7d6hlvix7fdy5iktfj3vynqfc4
*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. ...##
###
The Interlace Polynomial
[article]

2016
*
arXiv
*
pre-print

systems and

arXiv:1601.03003v1
fatcat:eqkt72dg3fgj5hmcxdh2dh2gti
*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*. ...##
###
Nullity and Loop Complementation for Delta-Matroids

2013
*
SIAM Journal on Discrete Mathematics
*

.,

doi:10.1137/110854692
fatcat:aoyrayvicbc7pbzl3grtuwocey
*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] . ...##
###
Canonical binary Δ-matroids
[article]

2022
*
arXiv
*
pre-print

We provide a canonical form for binary

arXiv:2203.04365v1
fatcat:b4y2frtq5ne6vhhitgs2z3uw3y
*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 }). ...##
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Isotropic Matroids I: Multimatroids and Neighborhoods

2016
*
Electronic Journal of Combinatorics
*

Several properties

doi:10.37236/5222
fatcat:htzpxbbxxrbv7j3av5prex5sge
*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. ...##
###
Author index to volume 181 (1998)

1998
*
Discrete Mathematics
*

Schw~irzler,

doi:10.1016/s0012-365x(97)82062-6
fatcat:sqdp43rjovam5az5gxnwxknhxq
*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) ...##
###
Isotropic matroids I: Multimatroids and neighborhoods
[article]

2016
*
arXiv
*
pre-print

Several properties

arXiv:1503.04406v5
fatcat:woqe5h2rpbdlniz2sjqwwdc4ka
*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. ...
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