Editing Graphic matroid

{{Use American English|date = January 2019}}
{{Short description|Matroid with graph forests as independent sets}}
[[File:Graphic matroid of C4.svg|thumb|upright=1.3|The '''graphic matroid''' of the [[cycle graph]] {{mvar|C{{sub|4}}}}, which is the [[uniform matroid]] <math>U{}^3_4</math>. More generally, the graphic matroid of {{mvar|C{{sub|n}}}} is <math>U{}^{n-1}_{n}</math>.<ref>
{{cite book
 | last = Welsh
 | first = D. J. A.
 | year = 2010
 | title = Matroid Theory
 | publisher = Courier Dover Publications
 | page = 10
 | isbn = 9780486474397
}}</ref>]]

In the mathematical theory of [[Matroid theory|matroid]]s, a '''graphic matroid''' (also called a '''cycle matroid''' or '''polygon matroid''') is a [[matroid]] whose independent sets are the [[tree (graph theory)|forests]] in a given finite [[undirected graph]]. The [[dual matroid]]s of graphic matroids are called '''co-graphic matroids''' or '''bond matroids'''.<ref>{{harvtxt|Tutte|1965}} uses a reversed terminology, in which he called bond matroids "graphic" and cycle matroids "co-graphic", but this has not been followed by later authors.</ref> A matroid that is both graphic and co-graphic is sometimes called a '''planar matroid''' (but this should not be confused with matroids of rank 3, which generalize planar point configurations); these are exactly the graphic matroids formed from [[planar graph]]s.

==Definition==
A [[matroid]] may be defined as a family of finite sets (called the "independent sets" of the matroid) that is closed under subsets and that satisfies the "exchange property": if sets <math>A</math> and <math>B</math> are both independent, and <math>A</math> is larger than <math>B</math>, then there is an element <math>x\in A\setminus B</math> such that <math>B\cup\{x\}</math> remains independent.  If <math>G</math> is an undirected graph, and <math>F</math> is the family of sets of edges that form forests in <math>G</math>, then <math>F</math> is clearly closed under subsets (removing edges from a forest leaves another forest). It also satisfies the exchange property: if <math>A</math> and <math>B</math> are both forests, and <math>A</math> has more edges than <math>B</math>, then it has fewer connected components, so by the [[pigeonhole principle]] there is a component <math>C</math> of <math>A</math> that contains vertices from two or more components of <math>B</math>. Along any path in <math>C</math> from a vertex in one component of <math>B</math> to a vertex of another component, there must be an edge with endpoints in two components, and this edge may be added to <math>B</math> to produce a forest with more edges. Thus, <math>F</math> forms the independent sets of a matroid, called the graphic matroid of <math>G</math> or <math>M(G)</math>. More generally, a matroid is called graphic whenever it is [[isomorphic]] to the graphic matroid of a graph, regardless of whether its elements are themselves edges in a graph.<ref name="tutte65"/>

The bases of a graphic matroid <math>M(G)</math> are the full [[spanning tree|spanning forests]] of <math>G</math>, and the circuits of <math>M(G)</math> are the [[cycle (graph theory)|simple cycles]] of <math>G</math>. The [[Matroid rank|rank]] in <math>M(G)</math> of a set <math>X</math> of edges of a graph <math>G</math> is <math>r(X)=n-c</math> where <math>n</math> is the number of vertices in the [[Glossary of graph theory#Subgraphs|subgraph]] formed by the edges in <math>X</math> and <math>c</math> is the number of connected components of the same subgraph.<ref name="tutte65"/> The corank of the graphic matroid is known as the [[circuit rank]] or cyclomatic number.

==The lattice of flats==
The [[matroid|closure]] <math>\operatorname{cl}(S)</math> of a set <math>S</math> of edges in <math>M(G)</math> is a [[matroid|flat]] consisting of the edges that are not independent of <math>S</math> (that is, the edges whose endpoints are connected to each other by a path in <math>S</math>). This flat may be identified with the partition of the vertices of <math>G</math> into the [[Connected component (graph theory)|connected components]] of the subgraph formed by <math>S</math>: Every set of edges having the same closure as <math>S</math> gives rise to the same partition of the vertices, and <math>\operatorname{cl}(S)</math> may be recovered from the partition of the vertices, as it consists of the edges whose endpoints both belong to the same set in the partition. In the [[geometric lattice|lattice of flats]] of this matroid, there is an order relation <math>x\le y</math> whenever the partition corresponding to flat&nbsp;<math>x</math> is a refinement of the partition corresponding to flat&nbsp;<math>y</math>.

In this aspect of graphic matroids, the graphic matroid for a [[complete graph]] <math>K_n</math> is particularly important, because it allows every possible partition of the vertex set to be formed as the set of connected components of some subgraph. Thus, the lattice of flats of the graphic matroid of <math>K_n</math> is naturally isomorphic to the [[partition of a set|lattice of partitions of an <math>n</math>-element set]]. Since the lattices of flats of matroids are exactly the [[geometric lattice]]s, this implies that the lattice of partitions is also geometric.<ref>{{citation|title=Lattice Theory|volume=25|series=Colloquium Publications|publisher=American Mathematical Society|first=Garrett|last=Birkhoff|authorlink=Garrett Birkhoff|edition=3rd|year=1995|isbn=9780821810255|page=95|url=https://books.google.com/books?id=0Y8d-MdtVwkC&pg=PA95}}.</ref> <!-- test -->

==Representation==
The graphic matroid of a graph <math>G</math> can be defined as the column matroid of any [[incidence matrix|oriented incidence matrix]] of <math>G</math>. Such a matrix has one row for each vertex, and one column for each edge. The column for edge <math>e</math> has <math>+1</math> in the row for one endpoint, <math>-1</math> in the row for the other endpoint, and <math>0</math> elsewhere; the choice of which endpoint to give which sign is arbitrary. The column matroid of this matrix has as its independent sets the linearly independent subsets of columns.

If a set of edges contains a cycle, then the corresponding columns (multiplied by <math>-1</math> if necessary to reorient the edges consistently around the cycle) sum to zero, and is not independent. Conversely, if a set of edges forms a forest, then by repeatedly removing leaves from this forest it can be shown by induction that the corresponding set of columns is independent. Therefore, the column matrix is isomorphic to <math>M(G)</math>.

This method of representing graphic matroids works regardless of the [[field (mathematics)|field]] over which the incidence is defined. Therefore, graphic matroids form a subset of the [[regular matroid]]s, matroids that have [[Matroid representation|representations]] over all possible fields.<ref name="tutte65"/>

The lattice of flats of a graphic matroid can also be realized as the lattice of a [[Arrangement of hyperplanes|hyperplane arrangement]], in fact as a subset of the [[Braid group|braid arrangement]], whose hyperplanes are the diagonals <math>H_{ij}=\{(x_1,\ldots,x_n) \in \mathbb{R}^n \mid x_i = x_j\}</math>. Namely, if the vertices of <math>G</math> are <math>v_1,\ldots,v_n,</math> we include the hyperplane <math>H_{ij}</math> whenever <math>e = v_iv_j</math> is an edge of <math>G</math>.

==Matroid connectivity==
A matroid is said to be connected if it is not the direct sum of two smaller matroids; that is, it is connected if and only if there do not exist two disjoint subsets of elements such that the rank function of the matroid equals the sum of the ranks in these separate subsets. Graphic matroids are connected if and only if the underlying graph is both [[connected graph|connected]] and [[k-vertex-connected graph|2-vertex-connected]].<ref name="tutte65"/>

==Minors and duality==
[[File:Nonisomorphic planar duals.svg|thumb|300px|Two different graphs (red) that are duals of the same planar graph (pale blue). Despite being non-isomorphic as graphs, they have isomorphic graphic matroids.]]
A matroid is graphic if and only if its [[Matroid minor|minors]] do not include any of five forbidden minors: the [[uniform matroid]] <math>U{}^2_4</math>, the [[Fano plane]] or its dual, or the duals of <math>M(K_5)</math> and <math>M(K_{3,3})</math> defined from the [[complete graph]] <math>K_5</math> and the [[complete bipartite graph]] <math>K_{3,3}</math>.<ref name="tutte65">{{citation
 | last = Tutte | first = W. T.
 | journal = Journal of Research of the National Bureau of Standards
 | mr = 0179781
 | pages = 1–47
 | title = Lectures on matroids
 | url = https://nvlpubs.nist.gov/nistpubs/jres/69B/jresv69Bn1-2p1_A1b.pdf
 | volume = 69B
 | year = 1965
 | doi=10.6028/jres.069b.001| doi-access = free
 }}. See in particular section 2.5, "Bond-matroid of a graph", pp. 5–6, section 5.6, "Graphic and co-graphic matroids", pp. 19–20, and section 9, "Graphic matroids", pp. 38–47.</ref><ref>{{citation
 | last = Seymour | first = P. D. | authorlink = Paul Seymour (mathematician)
 | doi = 10.1016/S0167-5060(08)70855-0
 | journal = Annals of Discrete Mathematics
 | mr = 597159
 | pages = 83–90
 | title = On Tutte's characterization of graphic matroids
 | volume = 8
 | year = 1980| isbn = 9780444861108 }}.</ref><ref>{{citation
 | last = Gerards | first = A. M. H.
 | doi = 10.1002/jgt.3190200311
 | issue = 3
 | journal = [[Journal of Graph Theory]]
 | mr = 1355434
 | pages = 351–359
 | title = On Tutte's characterization of graphic matroids—a graphic proof
 | volume = 20
 | year = 1995| s2cid = 31334681
 }}.</ref> The first three of these are the forbidden minors for the regular matroids,<ref>{{citation
 | last = Tutte | first = W. T. | authorlink = W. T. Tutte
 | journal = Transactions of the American Mathematical Society
 | mr = 0101526
 | pages = 144–174
 | title = A homotopy theorem for matroids. I, II
 | volume = 88
 | year = 1958
 | issue = 1 | doi=10.2307/1993244| jstor = 1993244 }}.</ref> and the duals of <math>M(K_5)</math> and <math>M(K_{3,3})</math> are regular but not graphic.

If a matroid is graphic, its dual (a "co-graphic matroid") cannot contain the duals of these five forbidden minors. Thus, the dual must also be regular, and cannot contain as minors the two graphic matroids <math>M(K_5)</math> and <math>M(K_{3,3})</math>.<ref name="tutte65"/>

Because of this characterization and [[Wagner's theorem]] characterizing the [[planar graph]]s as the graphs with no <math>K_5</math> or <math>K_{3,3}</math> [[graph minor]], it follows that a graphic matroid <math>M(G)</math> is co-graphic if and only if <math>G</math> is planar; this is [[Whitney's planarity criterion]]. If <math>G</math> is planar, the dual of <math>M(G)</math> is the graphic matroid of the [[dual graph]] of <math>G</math>. While <math>G</math> may have multiple dual graphs, their graphic matroids are all isomorphic.<ref name="tutte65"/>

==Algorithms==
A minimum weight basis of a graphic matroid is a [[minimum spanning tree]] (or minimum spanning forest, if the underlying graph is disconnected). Algorithms for computing minimum spanning trees have been intensively studied; it is known how to solve the problem in linear randomized expected time in a comparison model of computation,<ref>{{citation
 | last1 = Karger | first1 = David R. | author1-link = David Karger
 | last2 = Klein | first2 = Philip N.
 | last3 = Tarjan | first3 = Robert E. | author3-link = Robert Tarjan
 | doi = 10.1145/201019.201022
 | mr = 1409738
 | issue = 2
 | journal = [[Journal of the Association for Computing Machinery]]
 | pages = 321–328
 | title = A randomized linear-time algorithm to find minimum spanning trees
 | volume = 42
 | year = 1995| doi-access = free
 }}</ref> or in linear time in a model of computation in which the edge weights are small integers and bitwise operations are allowed on their binary representations.<ref>{{citation
 | last1 = Fredman | first1 = M. L. | author1-link = Michael Fredman
 | last2 = Willard | first2 = D. E. | author2-link = Dan Willard
 | doi = 10.1016/S0022-0000(05)80064-9
 | mr = 1279413
 | issue = 3
 | journal = [[Journal of Computer and System Sciences]]
 | pages = 533–551
 | title = Trans-dichotomous algorithms for minimum spanning trees and shortest paths
 | volume = 48
 | year = 1994| doi-access = free
 }}.</ref> The fastest known time bound that has been proven for a deterministic algorithm is slightly superlinear.<ref>{{citation
 | last = Chazelle | first = Bernard | authorlink = Bernard Chazelle
 | doi = 10.1145/355541.355562
 | mr = 1866456
 | issue = 6
 | journal = [[Journal of the Association for Computing Machinery]]
 | pages = 1028–1047
 | title = A minimum spanning tree algorithm with inverse-Ackermann type complexity
 | volume = 47
 | year = 2000| s2cid = 6276962 | doi-access = free
 }}.</ref>

Several authors have investigated algorithms for testing whether a given matroid is graphic.<ref>{{citation
 | last = Tutte | first = W. T. | authorlink = W. T. Tutte
 | journal = [[Proceedings of the American Mathematical Society]]
 | mr = 0117173
 | pages = 905–917
 | title = An algorithm for determining whether a given binary matroid is graphic.
 | volume = 11
 | year = 1960
 | issue = 6 | doi=10.2307/2034435| jstor = 2034435 }}.</ref><ref>{{citation
 | last1 = Bixby | first1 = Robert E.
 | last2 = Cunningham | first2 = William H.
 | doi = 10.1287/moor.5.3.321
 | issue = 3
 | journal = [[Mathematics of Operations Research]]
 | mr = 594849
 | pages = 321–357
 | title = Converting linear programs to network problems
 | volume = 5
 | year = 1980}}.</ref><ref>{{citation
 | last = Seymour | first = P. D. | authorlink = Paul Seymour (mathematician)
 | doi = 10.1007/BF02579179
 | issue = 1
 | journal = Combinatorica
 | mr = 602418
 | pages = 75–78
 | title = Recognizing graphic matroids
 | volume = 1
 | year = 1981| s2cid = 35579707 }}.</ref> For instance, an algorithm of {{harvtxt|Tutte|1960}} solves this problem when the input is known to be a [[binary matroid]]. {{harvtxt|Seymour|1981}} solves this problem for arbitrary matroids given access to the matroid only through an [[matroid oracle|independence oracle]], a subroutine that determines whether or not a given set is independent.

==Related classes of matroids==
Some classes of matroid have been defined from well-known families of graphs, by phrasing a characterization of these graphs in terms that make sense more generally for matroids. These include the [[bipartite matroid]]s, in which every circuit is even, and the [[Eulerian matroid]]s, which can be partitioned into disjoint circuits. A graphic matroid is bipartite if and only if it comes from a [[bipartite graph]] and a graphic matroid is Eulerian if and only if it comes from an [[Eulerian graph]]. Within the graphic matroids (and more generally within the [[binary matroid]]s) these two classes are dual: a graphic matroid is bipartite if and only if its [[dual matroid]] is Eulerian, and a graphic matroid is Eulerian if and only if its dual matroid is bipartite.<ref name="w69">{{citation
 | last = Welsh | first = D. J. A. | authorlink = Dominic Welsh
 | journal = [[Journal of Combinatorial Theory]]
 | mr = 0237368
 | pages = 375–377
 | title = Euler and bipartite matroids
 | volume = 6
 | year = 1969
 | issue = 4 | doi=10.1016/s0021-9800(69)80033-5
 | doi-access = free
 }}.</ref>

Graphic matroids are one-dimensional [[rigidity matroid]]s, matroids describing the degrees of freedom of structures of rigid beams that can rotate freely at the vertices where they meet. In one dimension, such a structure has a number of degrees of freedom equal to its number of connected components (the number of vertices minus the matroid rank) and in higher dimensions the number of degrees of freedom of a ''d''-dimensional structure with ''n'' vertices is ''dn'' minus the matroid rank. In two-dimensional rigidity matroids, the [[Laman graph]]s play the role that spanning trees play in graphic matroids, but the structure of rigidity matroids in dimensions greater than two is not well understood.<ref name="whiteley">{{citation
 | last = Whiteley | first = Walter | authorlink = Walter Whiteley
 | contribution = Some matroids from discrete applied geometry
 | doi = 10.1090/conm/197/02540
 | location = Providence, RI
 | mr = 1411692
 | pages = 171–311
 | publisher = American Mathematical Society
 | series = Contemporary Mathematics
 | title = Matroid theory (Seattle, WA, 1995)
 | volume = 197
 | year = 1996| doi-access = free
 | isbn = 978-0-8218-0508-4 }}.</ref>

==References==
{{reflist}}

[[Category:Matroid theory]]
[[Category:Planar graphs]]
[[Category:Graph connectivity]]
[[Category:Spanning tree]]