There are two ways to make a matroid smaller, one obvious and the other less so. Let M be a matroid, let H be a subset of its ground set G, and suppose that we want to build a matroid M0 on the smaller ground set G0 := G\H . The obvious
approach is to take, as the independent sets of M0, precisely those subsets of G0 that, viewed as subsets of G, are independent in M. This process is called deleting the elements of H , and the resulting matroid is written M\H . In the less obvious
approach, we choose some maximal independent subset I of H — it follows from the axioms that it doesn’t matter which we choose — and we let a subset X0of G0 be independent in M0precisely when the subset X := X0∪I of G is independent in M. This process is called contracting4the elements of H , and the resulting matroid is written M/H . A matroid M0is called a minor of a matroid M if we can get from
M to M0 by some sequence of deletions and contractions.
It is easy to see that deletions preserve representability. If H is any subset of the ground set of a matroid M, any representation of M can be converted into a representation of the deleted matroid M\H by simply forgetting about the points
to which the elements of H were mapped.
It is only slightly trickier to show that contractions also preserve representabil- ity. Let M be a representable matroid, let H be a subset of its ground set G, and fix some representationϕ of M, say in a projective space S of dimension n. The pointsϕ(H)to which the elements of H are mapped span a certain flat F in S, say of dimension k. Consider all flats of dimension k +1 in S that contain all of the points inϕ(H)and hence include F. Those(k +1)-flats themselves are the ‘points’ of a projective space S0of dimension n−k−1. For example, the lines in 3-space through a fixed point form a projective plane, while the planes in 3-space through a fixed line form a projective line.5 We can construct a representationϕ0of the contracted matroid M/H in the space S0by mapping each element e of G\H
to the(k+1)-flatϕ0(e):=Span(ϕ(H ∪ {e})). Typically, the span of the points in
ϕ(H ∪ {e})is a(k+1)-flat in S that includes F, so it is a point of S0. When e is
4Yet another term motivated by graph theory.
3.4. MINORS 37
a loop in the contracted matroid M/H , the pointϕ(e)lies in the flat F, and we set
ϕ0(e)to be the indeterminate point of S0.
Exercise 3.4-1 Let M be a matroid of rank r. The element e of M does not belong to any base just when e is a loop. In this case, show that deleting e and contracting
e produce the same matroid M\{e} = M/{e}, whose rank is r. On the other hand, suppose that the element e of M belongs to every base; such an element e is called a coloop. In this case also, show that deleting e and contracting e produce the same matroid M\{e} =M/{e}, but its rank is r−1. If the element e is neither a loop nor a coloop, show that deleting e gives a matroid M\{e}of rank r, while contracting
Chapter 4
The budget matroids
4.1
Initial examples
Each budget matroid Bb1,...,bk is associated with a partition b = b1+ · · · +bk of
a total budget b into k column budgets b1 through bk. To motivate the definition
of the budget matroids, let’s consider a couple of examples of partitions and their associated matroids.
We begin with the partition 3 = 1 +1+1. The budget matroid B1,1,1 turns out to be the Pappus matroid — the matroid whose representations are instances of the configuration that arises in Pappus’s Theorem.
As shown in Figure 4.1, let A1, A2, and A3be collinear points in the plane, let
B1, B2, and B3 also be collinear,1 and, whenever{i,j,k} = {1,2,3}, let Ci :=
AjBk∩ AkBj. Pappus’s Theorem tells us that the three intersection points C1, C2, and C3 of the opposite sides of the hexagon A1B2A3B1A2B3are also collinear. If
1I apologize for overusing the letter ‘B’: When a B has a single subscript, it denotes a point;
when it has multiple subscripts, it denotes a budget matroid.
A1 A2 A3 B1 B2 B 3 C1 C2 C3
Figure 4.1: An instance of the Pappus configuration.
40 CHAPTER 4. THE BUDGET MATROIDS A1 A2 A3 P1 P2 P3
Figure 4.2: An instance of the complete quadrilateral — as in Figure 2.2, but with the vertices(Bi)relabeled(Pi).
we arrange the nine points of the Pappus configuration in a matrix, like this,
1 1 1 A1 B1 C1 A2 B2 C2 A3 B3 C3 ,
we can describe the nine lines of the configuration as follows:
• The three points in a column are collinear. (But the three points in a row are not so constrained.)
• For each of the six possible ways of choosing one point from each row so that one is taken, also, from each column, the three points so chosen are collinear.
By the way, we often write the jthcolumn budget bj at the head of the jthcolumn,
as a reminder. In this example, all three column budgets are 1.
As a second example, consider the partition 3=2+1. A representation of the budget matroid B2,1is, it turns out, precisely a complete quadrilateral, as shown in Figure 4.2. If we arrange the six points of such a quadrilateral in the matrix
2 1 P1 A1 P2 A2 P3 A3 ,
its four lines can be described as follows:
• The three points in the A column are collinear, while the three points in the
P column are constrained only to be coplanar.
• For each of the three possible ways of choosing one point from each row so that two are taken from the P column and one from the A column, the three points so chosen are collinear.
4.1. INITIAL EXAMPLES 41
As for naming the points in a representation of a budget matroid, our standard convention will be to number the rows and to associate the letters A, B, C, and so forth with the columns, from left to right. But for matroids of the special form
Bm,1,1,...,1, where all of the column budgets except the first are ones, it is often con- venient to use a special letter for the first column and to start with A for the second column. Here, for the matroid B2,1, we have used the letter P for the first column, on the grounds that ‘P’ stands for ‘(co)planar’.
Note that we are failing to exploit all of the available symmetry in the two ex- amples above. The four lines of a complete quadrilateral play entirely symmetric roles. But to describe that quadrilateral as a representation of the matroid B2,1, we must choose one of its four lines to become the A column. In a similar way, to describe a Pappus configuration as a representation of the matroid B1,1,1, we must choose one of its three families of three disjoint lines to become the three columns. Failing to exploit a symmetry is often a mistake, but the unexploited symme- tries in those two cases are sporadic accidents. Consider a larger budget matroid, say the matroid B1,1,1,1associated with the partition 4=1+1+1+1:
1 1 1 1 A1 B1 C1 D1 A2 B2 C2 D2 A3 B3 C3 D3 A4 B4 C4 D4 .
A representation of B1,1,1,1consists of sixteen points in 3-space with the following two properties:
• The four points in each column are collinear.
• For each of the 24 possible ways of choosing one point from each row so that one is taken also from each column, the four points so chosen are coplanar. Since lines and planes are flats of different dimensions, there is no chance that some additional symmetry of the configuration might intermix the four lines required by the first property with the 24 planes required by the second. (For more about when such intermixing is possible and when it isn’t, see the analysis of the auto- morphisms of the budget matroids in Exercises 4.3-2 and 4.3-3.)
Exercise 4.1-1 (from Jorge Stolfi, and in preparation for Sections 10.3 and 10.4) If we write a point in the projective plane as a triple [w,x,y] of homogeneous
coordinates, verify that the nine points
A B C
1 [1,0,0] [0,1,0] [0,0,1] 2 [ p,q,q] [q,p,q] [q,q,p]
42 CHAPTER 4. THE BUDGET MATROIDS A1 A2 A3 B1 B2 B3 C1 C2 C3
Figure 4.3: A representation of the Pappus matroid B1,1,1in which all six permu- tations of the A, B, and C columns can be achieved via Euclidean symmetries.
form a Pappus configuration — that is, they represent the budget matroid B1,1,1— whenever the ratio p/q does not lie in{−2,−1,−12,0,1,∞}. What goes wrong for each of the six forbidden ratios?
Figure 4.3 shows an embedding in the Euclidean plane of the case p/q = 3. Note that the three lines A1A2A3, B1B2B3, and C1C2C3 are concurrent. Indeed, those lines will concur for any ratio p/q, at the point [1,1,1]. That concurrence is not required in a representation of the matroid B1,1,1 — indeed, it did not happen in Figure 4.1 — but it is allowed. It is allowed despite the fact that, for any subset
S of the nine points of the matroid, whenever the points in S are not required to be
mutually incident, they are required to be mutually skew.
These Pappus configurations are particularly symmetric: Any permutation of the three homogeneous coordinatesw, x , and y gives us a projective transforma- tion of the plane that is a symmetry of the configuration in which the rows are preserved and the columns are correspondingly permuted. Furthermore, when the configuration is embedded in the Euclidean plane as in Figure 4.3, those six projec- tive transformations become Euclidean transformations — in fact, the symmetries of the equilateral triangle4AiBiCi.
[Hint: When p/q = −2, the points A2, B2, and C2 all lie on the line with homogeneous coefficientsh1,1,1i— which, in Figure 4.3, is the line at infinity.]