@article{GARAMVOLGYI20241,
title = {Count and cofactor matroids of highly connected graphs},
journal = {Journal of Combinatorial Theory, Series B},
volume = {166},
pages = {1-29},
year = {2024},
issn = {0095-8956},
doi = {https://doi.org/10.1016/j.jctb.2023.12.004},
url = {https://www.sciencedirect.com/science/article/pii/S0095895623001120},
author = {Dániel Garamvölgyi and Tibor Jordán and Csaba Király},
keywords = {Count matroid, Cofactor matroid, Rigid graph, Vertical connectivity, Connected matroid},
abstract = {We consider two types of matroids defined on the edge set of a graph G: count matroids Mk,ℓ(G), in which independence is defined by a sparsity count involving the parameters k and ℓ, and the C21-cofactor matroid C(G), in which independence is defined by linear independence in the cofactor matrix of G. We show, for each pair (k,ℓ), that if G is sufficiently highly connected, then G−e has maximum rank for all e∈E(G), and the matroid Mk,ℓ(G) is connected. These results unify and extend several previous results, including theorems of Nash-Williams and Tutte (k=ℓ=1), and Lovász and Yemini (k=2,ℓ=3). We also prove that if G is highly connected, then the vertical connectivity of C(G) is also high. We use these results to generalize Whitney's celebrated result on the graphic matroid of G (which corresponds to M1,1(G)) to all count matroids and to the C21-cofactor matroid: if G is highly connected, depending on k and ℓ, then the count matroid Mk,ℓ(G) uniquely determines G; and similarly, if G is 14-connected, then its C21-cofactor matroid C(G) uniquely determines G. We also derive similar results for the t-fold union of the C21-cofactor matroid, and use them to prove that every 24-connected graph has a spanning tree T for which G−E(T) is 3-connected, which verifies a case of a conjecture of Kriesell.}
}
@article{geelen_computing_2018,
  title = {Computing {Girth} and {Cogirth} in {Perturbed} {Graphic} {Matroids}},
  volume = {38},
  issn = {0209-9683, 1439-6912},
  url = {http://link.springer.com/10.1007/s00493-016-3445-3},
  doi = {10.1007/s00493-016-3445-3},
  language = {en},
  number = {1},
  urldate = {2023-03-02},
  journal = {Combinatorica},
  author = {Geelen, Jim and Kapadia, Rohan},
  month = feb,
  year = {2018},
  pages = {167--191},
}

@article{Gu18,
  abstract = { Rigidity, arising in discrete geometry, is the property of a structure that does not flex. Laman provides a combinatorial characterization of rigid graphs in the Euclidean plane, and thus rigid graphs in the Euclidean plane have applications in graph theory. We discover a sufficient partition condition of packing spanning rigid subgraphs and spanning trees. As a corollary, we show that a simple graph \$G\$ contains a packing of \$k\$ spanning rigid subgraphs and l spanning trees if \$G\$ is \$(4k+2l)\$-edge-connected, and \$G-Z\$ is essentially \$(6k+2l - 2k|Z|)\$-edge-connected for every \$Z\subset V(G)\$. Thus every \$(4k+2l)\$-connected and essentially \$(6k+2l)\$-connected graph \$G\$ contains a packing of \$k\$ spanning rigid subgraphs and l spanning trees. Utilizing this, we show that every 6-connected and essentially 8-connected graph \$G\$ contains a spanning tree \$T\$ such that \$G-E(T)\$ is 2-connected. These improve some previous results. Sparse subgraph covering problems are also studied. },
  author = {Gu, Xiaofeng},
  doi = {10.1137/17M1134196},
  eprint = {https://doi.org/10.1137/17M1134196},
  journal = {SIAM Journal on Discrete Mathematics},
  number = {2},
  pages = {1305-1313},
  title = {Spanning Rigid Subgraph Packing and Sparse Subgraph Covering},
  url = {https://doi.org/10.1137/17M1134196},
  volume = {32},
  year = {2018},
  bdsk-url-1 = {https://doi.org/10.1137/17M1134196}}

@InProceedings{JordanKMM20,
author="Jord{\'a}n, Tibor
and Kobayashi, Yusuke
and Mahara, Ryoga
and Makino, Kazuhisa",
editor="G{\k{a}}sieniec, Leszek
and Klasing, Ralf
and Radzik, Tomasz",
title="The Steiner Problem for Count Matroids",
booktitle="Combinatorial Algorithms",
year="2020",
publisher="Springer International Publishing",
address="Cham",
pages="330--342",
abstract="We introduce and study a generalization of the well-known Steiner tree problem to count matroids. In the count matroid {\$}{\$}{\backslash}mathcal{\{}M{\}}{\_}{\{}k,l{\}}(G){\$}{\$}, defined on the edge set of a graph {\$}{\$}G=(V,E){\$}{\$}, a set {\$}{\$}F{\backslash}subseteq E{\$}{\$} is independent if every vertex set {\$}{\$}X{\backslash}subseteq V{\$}{\$} spans at most {\$}{\$}k|X|-l{\$}{\$} edges of F. The graph is called (k, l)-tight if its edge set is independent in {\$}{\$}{\backslash}mathcal{\{}M{\}}{\_}{\{}k,l{\}}(G){\$}{\$} and {\$}{\$}|E|=k|V|-l{\$}{\$} holds.",
isbn="978-3-030-48966-3"
}


@article{LeeS08,
  title = {Pebble game algorithms and sparse graphs},
  volume = {308},
  issn = {0012365X},
  doi = {10.1016/j.disc.2007.07.104},
  abstract = {A multi-graph G on n vertices is (k, ℓ)-sparse if every subset of n′ ≤ n vertices spans at most kn′ - ℓ edges. G is tight if, in addition, it has exactly kn - ℓ edges. For integer values k and ℓ ∈ [0, 2 k), we characterize the (k, ℓ)-sparse graphs via a family of simple, elegant and efficient algorithms called the (k, ℓ)-pebble games. © 2007 Elsevier B.V. All rights reserved.},
  number = {8},
  journal = {Discrete Mathematics},
  author = {Lee, Audrey and Streinu, Ileana},
  year = {2008},
  note = {arXiv: math/0702129},
  keywords = {Circuit, Henneberg sequence, Matroid, Pebble game, Rigidity, Sparse graph},
  pages = {1425--1437},
  file = {PDF:/Users/chaoxu/Zotero/storage/NP5ACYFI/2008-Pebble_game_algorithms_and_sparse_graphs.pdf:application/pdf},
}

@article{StreinuT09,
  title = {Sparsity-certifying {Graph} {Decompositions}},
  volume = {25},
  issn = {1435-5914},
  url = {https://doi.org/10.1007/s00373-008-0834-4},
  doi = {10.1007/s00373-008-0834-4},
  abstract = {We describe a new algorithm, the (k, ℓ)-pebble game with colors, and use it to obtain a characterization of the family of (k, ℓ)-sparse graphs and algorithmic solutions to a family of problems concerning tree decompositions of graphs. Special instances of sparse graphs appear in rigidity theory and have received increased attention in recent years. In particular, our colored pebbles generalize and strengthen the previous results of Lee and Streinu [12] and give a new proof of the Tutte-Nash-Williams characterization of arboricity. We also present a new decomposition that certifies sparsity based on the (k, ℓ)-pebble game with colors. Our work also exposes connections between pebble game algorithms and previous sparse graph algorithms by Gabow [5], Gabow and Westermann [6] and Hendrickson [9].},
  number = {2},
  journal = {Graphs and Combinatorics},
  author = {Streinu, Ileana and Theran, Louis},
  month = may,
  year = {2009},
  pages = {219--238},
}


@article{Servatius91,
  abstract = {We give a characterization of the dual of the 2-dimensional generic rigidity matroid R(G) of a graph G and derive necessary and sufficient conditions for a connected matroid to be the rigidity matroid of a birigid graph.},
  author = {Brigitte Servatius},
  doi = {https://doi.org/10.1016/0095-8956(91)90056-P},
  issn = {0095-8956},
  journal = {Journal of Combinatorial Theory, Series B},
  number = {1},
  pages = {106-113},
  title = {On the two-dimensional generic rigidity matroid and its dual},
  url = {https://www.sciencedirect.com/science/article/pii/009589569190056P},
  volume = {53},
  year = {1991},
  bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/009589569190056P},
  bdsk-url-2 = {https://doi.org/10.1016/0095-8956(91)90056-P}}

@PHDTHESIS {Fahad15,
    author = "Fahad, P",
    title  = "Dynamic Programming using Representative Families",
    school = "HOMI BHABHA NATIONAL INSTITUTE",
    year   = "2015",
    month  = "jul"
}

@article {BaileyNS14,
    AUTHOR = {Bailey, Robert F. and Newman, Mike and Stevens, Brett},
     TITLE = {A note on packing spanning trees in graphs and bases in
              matroids},
   JOURNAL = {Australas. J. Combin.},
  FJOURNAL = {The Australasian Journal of Combinatorics},
    VOLUME = {59},
      YEAR = {2014},
     PAGES = {24--38},
      ISSN = {1034-4942}
}
@article{ChekuriGN06,
  author = {Chandra Chekuri and Sudipto Guha and Joseph (Seffi) Naor},
  title = {The Steiner k-Cut Problem},
  journal = {SIAM Journal on Discrete Mathematics},
  volume = {20},
  number = {1},
  pages = {261-271},
  year = {2006},
  doi = {10.1137/S0895480104445095},
}
@article{Vardy97,
 author = {Alexander Vardy},
 title = {The Intractability of Computing the Minimum Distance of a Code},
 journal = {IEEE Trans. Inf. Theor.},
 volume = {43},
 number = {6},
 month = {Nov},
 year = {1997},
 issn = {0018-9448},
 pages = {1757--1766},
 numpages = {10},
 doi = {10.1109/18.641542},
 acmid = {2265234},
 publisher = {IEEE Press},
 address = {Piscataway, NJ, USA},
}
@InProceedings{FominGLS17,
  author ={Fedor V. Fomin and Petr A. Golovach and Daniel Lokshtanov and Saket Saurabh},
  title ={{Covering Vectors by Spaces: Regular Matroids}},
  booktitle ={44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
  pages ={56:1--56:15},
  series ={Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN ={978-3-95977-041-5},
  ISSN ={1868-8969},
  year ={2017},
  volume ={80},
  editor ={Ioannis Chatzigiannakis and Piotr Indyk and Fabian Kuhn and Anca Muscholl},
  publisher ={Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address ={Dagstuhl, Germany},
  doi ={10.4230/LIPIcs.ICALP.2017.56}
}
@book{Oxley06,
  title={Matroid Theory},
  author={Oxley, James G.},
  isbn={9780199202508},
  series={Oxford graduate texts in mathematics},
  year={2006},
  publisher={Oxford University Press}
}
@Inbook{PanolanRS15,
  author="Panolan, Fahad
  and Ramanujan, M. S.
  and Saurabh, Saket",
  title="On the Parameterized Complexity of Girth and Connectivity Problems on Linear Matroids",
  bookTitle="Algorithms and Data Structures: 14th International Symposium, WADS 2015, Victoria, BC, Canada, August 5-7, 2015. Proceedings",
  year="2015",
  publisher="Springer International Publishing",
  pages="566--577",
  isbn="978-3-319-21840-3",
  doi="10.1007/978-3-319-21840-3_47",
}
@techreport{Kiraly09,
  AUTHOR  = {Kir{\'a}ly, Tam{\'a}s},
  TITLE = {Computing the minimum cut in hypergraphic matroids},
  NOTE  = {{\tt www.cs.elte.hu/egres}},
  INSTITUTION = {Egerv{\'a}ry Research Group, Budapest},
  YEAR  = {2009},
  NUMBER  = {QP-2009-05}
}
@article{JoretV15,
  TITLE = {{Reducing the rank of a matroid}},
  AUTHOR = {Joret, Gwenaël and Vetta, Adrian},
  JOURNAL = {{Discrete Mathematics \& Theoretical Computer Science}},
  VOLUME = {{Vol. 17 no.2}},
  YEAR = {2015},
  MONTH = Sep
}
@article{karger_minimum_2000,
	author = {Karger, David R.},
	title = {Minimum cuts in near-linear time},
	journal = {Journal of the ACM},
	volume = {47},
	number = {1},
	pages = {46--76},
	year = {2000},
	doi = {10.1145/331605.331608},
}

@article{Karger98,
    title = {Random sampling and greedy sparsification for matroid optimization problems},
    volume = {82},
    issn = {0025-5610},
    doi = {10.1007/BF01585865},
    abstract = {Random sampling is a powerful tool for gathering information about a{\textbackslash}ngroup by considering only a small part of it. We discuss some broadly{\textbackslash}napplicable paradigms for using random sampling in combinatorial{\textbackslash}noptimization, and demonstrate the effectiveness of these paradigms for{\textbackslash}ntwo optimization problems on matroids: finding an optimum matroid basis{\textbackslash}nand packing disjoint matroid bases. Application of these ideas to the{\textbackslash}ngraphic matroid led to fast algorithms for minimum spanning trees and{\textbackslash}nminimum cuts. An optimum matroid basis is typically found by a greedy{\textbackslash}nalgorithm that grows an independent set into an optimum bash; one{\textbackslash}nelement at a time. This continuous change in the independent set can{\textbackslash}nmake it hard to perform the independence tests needed by the greedy{\textbackslash}nalgorithm. We simplify matters by using sampling to reduce the problem{\textbackslash}nof finding an optimum matroid basis to the problem of verifying that a{\textbackslash}ngiven fixed basis is optimum, showing that the two problems can be{\textbackslash}nsolved in roughly the same time. Another application of sampling is to{\textbackslash}npacking matroid bases, also known as matroid partitioning. Sampling{\textbackslash}nreduces the number of bases that must be packed. We combine sampling{\textbackslash}nwith a greedy packing strategy that reduces the size of the matroid.{\textbackslash}nTogether, these techniques give accelerated packing algorithms. We give{\textbackslash}nparticular attention to the problem of packing spanning trees in graphs,{\textbackslash}nwhich has applications in network reliability analysis. Our results can{\textbackslash}nbe seen as generalizing certain results from random graph theory.The{\textbackslash}ntechniques :have also been effective for other packing problems. (C){\textbackslash}n1998 The Mathematical Programming Society, Inc. Published by Elsevier{\textbackslash}nScience B.V.},
    number = {1-2},
    journal = {Mathematical Programming},
    author = {Karger, David R},
    year = {1998},
    keywords = {greedy algorithm, matroid basis, random sampling},
    pages = {41--81},
    file = {PDF:/Users/chaoxu/Zotero/storage/FN258PYE/Karger - 1998 - Random sampling and greedy sparsification for matroid optimization problems.pdf:application/pdf},
}
@article{GurjarTV17,
  author    = {Rohit Gurjar and
               Thomas Thierauf and
               Nisheeth K. Vishnoi},
  title     = {Isolating a Vertex via Lattices: Polytopes with Totally Unimodular
               Faces},
  journal   = {CoRR},
  volume    = {abs/1708.02222},
  year      = {2017},
  url       = {http://arxiv.org/abs/1708.02222},
  timestamp = {Tue, 05 Sep 2017 10:03:46 +0200},
  biburl    = {http://dblp.org/rec/bib/journals/corr/abs-1708-02222},
  bibsource = {dblp computer science bibliography, http://dblp.org}
}

@book{schrijver_combinatorial_2003,
	address = {Berlin Heidelberg},
	series = {Algorithms and combinatorics},
	title = {Combinatorial optimization: polyhedra and efficiency},
	isbn = {978-3-540-44389-6},
	shorttitle = {Combinatorial optimization},
	language = {en},
	number = {24},
	publisher = {Springer},
	author = {Schrijver, Alexander},
	year = {2003},
	file = {Schrijver - 2003 - Combinatorial optimization polyhedra and efficien.pdf:/Users/congyu/Zotero/storage/8369KL3F/Schrijver - 2003 - Combinatorial optimization polyhedra and efficien.pdf:application/pdf},
}

@article{chekuri_lp_2020,
	title = {{LP} {Relaxation} and {Tree} {Packing} for {Minimum} \$k\$-{Cut}},
	volume = {34},
	issn = {0895-4801, 1095-7146},
	url = {https://epubs.siam.org/doi/10.1137/19M1299359},
	doi = {10.1137/19M1299359},
	abstract = {Karger used spanning tree packings [D. R. Karger, J. ACM, 47 (2000), pp. 46-76] to derive a near linear-time randomized algorithm for the global minimum cut problem as well as a bound on the number of approximate minimum cuts. This is a different approach from his well-known random contraction algorithm [D. R. Karger, Random Sampling in Graph Optimization Problems, Ph.D. thesis, Stanford University, Stanford, CA, 1995, D. R. Karger and C. Stein, J. ACM, 43 (1996), pp. 601--640]. Thorup developed a fast deterministic algorithm for the minimum k-cut problem via greedy recursive tree packings [M. Thorup, Minimum k-way cuts via deterministic greedy tree packing, in Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, ACM, 2008, pp. 159--166]. In this paper we revisit properties of an LP relaxation for k-Cut proposed by Naor and Rabani [Tree packing and approximating k-cuts, in Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms, Vol. 103, SIAM, Philadelphia, 2001, pp. 26--27], and analyzed in [C. Chekuri, S. Guha, and J. Naor, SIAM J. Discrete Math., 20 (2006), pp. 261--271]. We show that the dual of the LP yields a tree packing that, when combined with an upper bound on the integrality gap for the LP, easily and transparently extends Karger's analysis for mincut to the k-cut problem. In addition to the simplicity of the algorithm and its analysis, this allows us to improve the running time of Thorup's algorithm by a factor of n. We also improve the bound on the number of {\textbackslash}alpha -approximate k-cuts. Second, we give a simple proof that the integrality gap of the LP is 2(1  -  1/n). Third, we show that an optimum solution to the LP relaxation, for all values of k, is fully determined by the principal sequence of partitions of the input graph. This allows us to relate the LP relaxation to the Lagrangean relaxation approach of Barahona [Oper. Res. Lett., 26 (2000), pp. 99--105] and Ravi and Sinha [European J. Oper. Res., 186 (2008), pp. 77--90]; it also shows that the idealized recursive tree packing considered by Thorup gives an optimum dual solution to the LP.},
	language = {en},
	number = {2},
	urldate = {2022-04-10},
	journal = {SIAM Journal on Discrete Mathematics},
	author = {Chekuri, Chandra and Quanrud, Kent and Xu, Chao},
	month = jan,
	year = {2020},
	keywords = {Approximation, K-cut, Minimum cut, Tree packing},
	pages = {1334--1353},
	file = {Chekuri et al. - 2020 - LP Relaxation and Tree Packing for Minimum \$k\$-Cut.pdf:/Users/congyu/Zotero/storage/XDUPHUTC/Chekuri et al. - 2020 - LP Relaxation and Tree Packing for Minimum \$k\$-Cut.pdf:application/pdf},
}

@inproceedings{boros2003algorithms,
  title={Algorithms for enumerating circuits in matroids},
  author={Boros, Endre and Elbassioni, Khaled and Gurvich, Vladimir and Khachiyan, Leonid},
  booktitle={International Symposium on Algorithms and Computation},
  pages={485--494},
  year={2003},
  organization={Springer}
}

@incollection{whiteley_matroids_1996,
	address = {Providence, Rhode Island},
	title = {Some matroids from discrete applied geometry},
	volume = {197},
	isbn = {978-0-8218-0508-4 978-0-8218-7788-3},
	url = {http://www.ams.org/conm/197/},
	abstract = {We present an array of matroids drawn from three sources in discrete applied geometry: (i) static (or ﬁrst-order) rigidity of frameworks and higher skeletal rigidity; (ii) parallel drawings (or equivalently polyhedral pictures); and (iii) Crr-1-cofactors abstracted from multivariate splines in all dimensions. The strong analogies (sometimes isomorphisms) between generic rigidity matroids and generic cofactor matroids is one central theme of the chapter. We emphasize matroidal results for the combinatorial ‘generic’ situations, with geometric techniques used when they contribute combinatorial insights. A second basic theme is the analysis of represented matroids using the duality of row and column dependencies of the representing matrix (generalizing statics and kinematics in rigidity).},
	language = {en},
	urldate = {2024-06-11},
	booktitle = {Contemporary {Mathematics}},
	publisher = {American Mathematical Society},
	author = {Whiteley, Walter},
	editor = {Bonin, Joseph E. and Oxley, James G. and Servatius, Brigitte},
	year = {1996},
	doi = {10.1090/conm/197/02540},
	pages = {171--311},
	file = {Whiteley - 1996 - Some matroids from discrete applied geometry.pdf:/Users/congyu/Zotero/storage/VG5DFCBQ/Whiteley - 1996 - Some matroids from discrete applied geometry.pdf:application/pdf},
}

@article{pym_submodular_1970,
	title = {Submodular functions and independence structures},
	volume = {30},
	copyright = {https://www.elsevier.com/tdm/userlicense/1.0/},
	issn = {0022247X},
	url = {https://linkinghub.elsevier.com/retrieve/pii/0022247X70901800},
	doi = {10.1016/0022-247X(70)90180-0},
	language = {en},
	number = {1},
	urldate = {2024-06-14},
	journal = {Journal of Mathematical Analysis and Applications},
	author = {Pym, J.S and Perfect, Hazel},
	month = apr,
	year = {1970},
	pages = {1--31},
	file = {Pym and Perfect - 1970 - Submodular functions and independence structures.pdf:/Users/congyu/Zotero/storage/M85CHJF6/Pym and Perfect - 1970 - Submodular functions and independence structures.pdf:application/pdf},
}

@book{frank_connections_2011,
	title = {Connections in {Combinatorial} {Optimization}},
	isbn = {978-0-19-920527-1},
	url = {http://scholar.google.com/scholar?hl=en&btnG=Search&q=intitle:Connections+in+Combinatorial+Optimization#0},
	urldate = {2014-07-17},
	publisher = {Oxford University Press},
	author = {Frank, András},
	year = {2011},
	note = {Publication Title: Oxford Lecture Series in Mathematics and Its Applications},
	file = {PDF:/Users/congyu/Zotero/storage/7WP6YL2K/2011-Connections_in_Combinatorial_Optimization.pdf:application/pdf},
}

@article{catlin_fractional_1992,
	title = {Fractional arboricity, strength, and principal partitions in graphs and matroids},
	volume = {40},
	copyright = {https://www.elsevier.com/tdm/userlicense/1.0/},
	issn = {0166218X},
	url = {https://linkinghub.elsevier.com/retrieve/pii/0166218X9290002R},
	doi = {10.1016/0166-218X(92)90002-R},
	abstract = {Catlin, P.A., J.W. Grossman, A.M. Hobbs and H.-J. Lai, Fractional arboricity, strength, and principal partitions in graphs and matroids, Discrete Applied Mathematics 40 (1992) 285-302.},
	language = {en},
	number = {3},
	urldate = {2024-04-29},
	journal = {Discrete Applied Mathematics},
	author = {Catlin, Paul A. and Grossman, Jerrold W. and Hobbs, Arthur M. and Lai, Hong-Jian},
	month = dec,
	year = {1992},
	pages = {285--302},
}

@article{haas_characterizations_2002,
	title = {Characterizations of {Arboricity} of {Graphs}},
	volume = {63},
	abstract = {The aim of this paper is to give several characterizations for the following two classes of graphs: (i) graphs for which adding any l edges produces a graph which is decomposible into k spanning trees and (ii) graphs for which adding some l edges produces a graph which is decomposible into k spanning trees.},
	journal = {Ars Comb.},
	author = {Haas, Ruth},
	month = apr,
	year = {2002},
	keywords = {base packing, sparsity},
	file = {Full Text PDF:/Users/congyu/Zotero/storage/HJJNU6UK/Haas - 2002 - Characterizations of Arboricity of Graphs.pdf:application/pdf},
}

@article{jackson_generic_2010,
	series = {Combinatorics and {Geometry}},
	title = {The generic rank of body--bar-and-hinge frameworks},
	volume = {31},
	issn = {0195-6698},
	url = {https://www.sciencedirect.com/science/article/pii/S0195669809000973},
	doi = {10.1016/j.ejc.2009.03.030},
	abstract = {Tay [T.S. Tay, Rigidity of multi-graphs I Linking Bodies in n-space, J. Combin. Theory B 26 (1984) 95--112] characterized the multigraphs which can be realized as infinitesimally rigid d-dimensional body-and-bar frameworks. Subsequently, Tay [T.S. Tay, Linking (n−2)-dimensional panels in n-space II: (n−2,2)-frameworks and body and hinge structures, Graphs Combin. 5 (1989) 245--273] and Whiteley [W. Whiteley, The union of matroids and the rigidity of frameworks, SIAM J. Discrete Math. 1 (2) (1988) 237--255] independently characterized the multigraphs which can be realized as infinitesimally rigid d-dimensional body-and-hinge frameworks. We adapt Whiteley’s proof technique to characterize the multigraphs which can be realized as infinitesimally rigid d-dimensional body--bar-and-hinge frameworks. More importantly, we obtain a sufficient condition for a multigraph to be realized as an infinitesimally rigid d-dimensional body-and-hinge framework in which all hinges lie in the same hyperplane. This result is related to a long-standing conjecture of Tay and Whiteley [T.S. Tay, W. Whiteley, Recent advances in the generic rigidity of structures, Structural Topology 9 (1984) 31--38] which would characterize when a multigraph can be realized as an infinitesimally rigid d-dimensional body-and-hinge framework in which all the hinges incident to each body lie in a common hyperplane. As a corollary we deduce that if a graph G has three spanning trees which use each edge of G at most twice, then its square can be realized as an infinitesimally rigid three-dimensional bar-and-joint framework.},
	number = {2},
	urldate = {2026-03-12},
	journal = {European Journal of Combinatorics},
	author = {Jackson, Bill and Jord{\'a}n, Tibor},
	month = feb,
	year = {2010},
	pages = {574--588},
}

@article{Nash-Williams_1961,
	title = {Edge-Disjoint Spanning Trees of Finite Graphs},
	volume = {s1-36},
	rights = {http://doi.wiley.com/10.1002/tdm_license_1.1},
	issn = {00246107},
	url = {http://doi.wiley.com/10.1112/jlms/s1-36.1.445},
	doi = {10.1112/jlms/s1-36.1.445},
	number = {1},
	journal = {Journal of the London Mathematical Society},
	author = {Nash-Williams, C. St.J. A.},
	year = {1961},
	pages = {445--450},
	language = {en},
}

@article{fujishige_lexicographically_1980,
	title = {Lexicographically {Optimal} {Base} of a {Polymatroid} with {Respect} to a {Weight} {Vector}},
	volume = {5},
	issn = {0364-765X},
	url = {https://doi.org/10.1287/moor.5.2.186},
	doi = {10.1287/moor.5.2.186},
	number = {2},
	journal = {Mathematics of Operations Research},
	author = {Fujishige, Satoru},
	year = {1980},
	pages = {186--196},
}

@incollection{fujishige_principal_2009,
	title = {Theory of {Principal} {Partitions} {Revisited}},
	booktitle = {Research {Trends} in {Combinatorial} {Optimization}},
	publisher = {Springer Berlin Heidelberg},
	address = {Berlin, Heidelberg},
	author = {Fujishige, Satoru},
	editor = {Cook, William J. and Lov{\'a}sz, L{\'a}szl{\'o} and Vygen, Jens},
	year = {2009},
	pages = {127--162},
	doi = {10.1007/978-3-540-76796-1_7},
}

@article{truong_modulus_2025,
	title = {Modulus for bases of matroids},
	volume = {348},
	url = {https://doi.org/10.1016/j.disc.2025.114395},
	doi = {10.1016/j.disc.2025.114395},
	number = {5},
	journal = {Discrete Mathematics},
	author = {Truong, Huy and Poggi-Corradini, Pietro},
	year = {2025},
	pages = {114395},
}

@article{zaslavsky_biased_1989,
	author = {Zaslavsky, Thomas},
	title = {Biased graphs. {I}. {B}ias, balance, and gains},
	journal = {Journal of Combinatorial Theory, Series B},
	volume = {47},
	number = {1},
	pages = {32--52},
	year = {1989},
	doi = {10.1016/0095-8956(89)90063-4},
}

@article{zaslavsky_biased_1991,
	author = {Zaslavsky, Thomas},
	title = {Biased graphs. {II}. {T}he three matroids},
	journal = {Journal of Combinatorial Theory, Series B},
	volume = {51},
	number = {1},
	pages = {46--72},
	year = {1991},
	doi = {10.1016/0095-8956(91)90005-5},
}

@article{dejesus_kcircular_2017,
	author = {De Jes\'{u}s, Jos\'{e} F. and Kelmans, Alexander K.},
	title = {\(k\)-circular matroids of graphs},
	journal = {Discrete Applied Mathematics},
	volume = {225},
	pages = {33--50},
	year = {2017},
	doi = {10.1016/j.dam.2017.02.026},
}

@incollection{dinits_structure_1976,
	author = {Dinits, E. A. and Karzanov, A. V. and Lomonosov, M. V.},
	title = {On the structure of a family of minimal weighted cuts in a graph},
	booktitle = {Studies in Discrete Optimization},
	editor = {Fridman, A. A.},
	publisher = {Nauka},
	address = {Moscow},
	year = {1976},
	pages = {290--306},
	note = {In Russian},
}

@article{goldschmidt_polynomial_1994,
	author = {Goldschmidt, Olivier and Hochbaum, Dorit S.},
	title = {A Polynomial Algorithm for the $k$-Cut Problem for Fixed $k$},
	journal = {Mathematics of Operations Research},
	volume = {19},
	number = {1},
	pages = {24--37},
	year = {1994},
	doi = {10.1287/moor.19.1.24},
}

@article{berczi_blocking_2018,
	author = {B\'{e}rczi, Krist\'{o}f and Bern\'{a}th, Attila and Kir\'{a}ly, Tam\'{a}s and Pap, Gyula},
	title = {Blocking optimal structures},
	journal = {Discrete Mathematics},
	volume = {341},
	number = {7},
	pages = {1864--1872},
	year = {2018},
	doi = {10.1016/j.disc.2018.03.024},
}

@inproceedings{eren_rigidity_2004,
	author = {Eren, Tolga and Goldenberg, D. K. and Whiteley, Walter and Yang, Yang Richard and Belhumeur, Peter N. and Anderson, Brian D. O. and Morse, A. Stephen},
	title = {Rigidity, computation, and randomization in network localization},
	booktitle = {Proceedings of IEEE INFOCOM},
	year = {2004},
	doi = {10.1109/INFCOM.2004.1354686},
}

@inproceedings{yu_agent_2008,
	author = {Yu, Changbin and Anderson, Brian D. O.},
	title = {Agent and link redundancy for autonomous formations},
	booktitle = {Proceedings of the 17th World Congress, International Federation of Automatic Control},
	year = {2008},
	doi = {10.3182/20080706-5-KR-1001.0554},
}

@article{jacobs_protein_2001,
	author = {Jacobs, Donald J. and Rader, A. J. and Kuhn, Leslie A. and Thorpe, M. F.},
	title = {Protein flexibility predictions using graph theory},
	journal = {Proteins: Structure, Function, and Bioinformatics},
	volume = {44},
	number = {2},
	pages = {150--165},
	year = {2001},
	doi = {10.1002/prot.1081},
}

@inproceedings{thorup_minimum_2008,
	author = {Thorup, Mikkel},
	title = {Minimum $k$-way cuts via deterministic greedy tree packing},
	booktitle = {Proceedings of the 40th Annual ACM Symposium on Theory of Computing},
	pages = {159--166},
	year = {2008},
	doi = {10.1145/1374376.1374402},
}

@article{gupta_optimal_2022,
	author = {Gupta, Anupam and Harris, David G. and Lee, Euiwoong and Li, Jason},
	title = {Optimal Bounds for the $k$-cut Problem},
	journal = {Journal of the ACM},
	volume = {69},
	number = {1},
	pages = {1--18},
	year = {2022},
	doi = {10.1145/3478018},
}

@techreport{jordan_combinatorial_2014,
	address = {Budapest, Hungary},
	title = {Combinatorial rigidity: graphs and matroids in the theory of rigid frameworks},
	issn = {1587–4451},
	url = {http://www.cs.elte.hu/egres},
	language = {en},
	number = {TR-2014-12},
	institution = {Egerváry Research Group},
	author = {Jordán, Tibor},
	month = sep,
	year = {2014},
	file = {Jordan - COMBINATORIAL RIGIDITY GRAPHS AND MATROIDS IN THE.pdf:/Users/congyu/Zotero/storage/RRMXGIDT/Jordan - COMBINATORIAL RIGIDITY GRAPHS AND MATROIDS IN THE.pdf:application/pdf},
}
@article{graver_rigidity_1991,
	title = {Rigidity {Matroids}},
	volume = {4},
	issn = {0895-4801, 1095-7146},
	url = {http://epubs.siam.org/doi/10.1137/0404032},
	doi = {10.1137/0404032},
	abstract = {This paper begins with a short discussion of the general principles of Rigidity Theory. The main interest is the combinatorial part ofthis subject: generic rigidity. While generic rigidity has several combinatorial characterizations in dimensions one and two, these characterizations have not been able to be extended to characterizations of generic rigidity in higher dimensions. In fact, no "purely combinatorial" characterization is presently known for generic rigidity in dimensions three and up. The concept of an abstract rigidity matroid is introduced and, in the context of matroid theory, the present status of the characterization problem is discussed.},
	language = {en},
	number = {3},
	urldate = {2025-10-10},
	journal = {SIAM Journal on Discrete Mathematics},
	author = {Graver, Jack E.},
	month = aug,
	year = {1991},
	pages = {355--368},
	file = {PDF:/Users/congyu/Zotero/storage/GPKMUI3Y/Graver - 1991 - Rigidity Matroids.pdf:application/pdf},
}

