Extremal Matroids and Growth Rates

Guest post by Sandra Kingan

Growth rates are a popular topic on Matroid Union. Peter Nelson has written five posts on it: Growth Rates IIIIIIIVV.  Irene Pivotto has also written a post on it.

Following James Oxley’s book [Oxley, 2012] , the growth rate function of a minor-closed class $\mathcal M$, denoted by $h_{\mathcal M}(r)$, is defined as the maximum number of elements in a simple rank-$r$ matroid in $\mathcal M$ if the number is finite and infinity otherwise (p. 570). A simple matroid is extremal in a minor-closed class $\mu$ of matroid if $M\in \mu$, but $\mu$ contains no simple single-element extension of $M$ that has the same rank as $M$ (p. 572).

For example, if $\mathcal M$ is the class of graphs, then the complete graph of rank-$r$ denoted by $K_{r+1}$, for $r\ge 1$ is the rank-$r$ extremal matroid. The growth rate function is the size of $K_{r+1}$ which is $\frac {r(r+1)}{2}$.  On the other hand if $\mathcal M$ is the subclass of planar graphs, then we don’t know precisely the extremal planar graphs, but we do know the growth rate function. A straightforward graph theoretic argument (found in any graph theory textbook) shows that the number of edges of a “maximal” planar graph is $3r-3$. (In graph theory a maximal planar graph is defined as one to which if you add one more edge it becomes non-planar.)

Similarly, if $\mathcal M$ is the class of binary matroids, then the rank-$r$ projective geometry $PG(r-1, 2)$, for $r\ge 2$, is the rank-$r$ extremal matroid. The growth rate function is the size of $PG(r-1, 2)$ which is $2^r-1$.

Within the class of binary matroids lies the class of cographic matroids (matroids whose duals are graphic). Again we don’t know precisely the extremal cographic matroids, but we can prove that the growth rate function is $3r-3$ (as shown by Joseph Kung and further explained further in Pivotto’s post). Also in her post is the growth rate of series-parallel networks. Series-parallel networks are formed by starting with a single edge or loop and repeatedly adding edges in parallel or edges in series (turning one edge into two edges by putting a vertex on the edge). The growth rate of series parallel networks is $2r+1$. See [Oxley Section 5.4] and [Oxley 1987, Section 3] for the explanation.

The first extremal result is generally attributed to Paul Turán. The class he considered was the class of graphs with no $K_{s+1}$-subgraph. (Mantel proved the $K_3$ case earlier.) Turán proved that the rank $r$ extremal graph is a specific type of complete s-partite graph (see Wikipedia). The growth rate function is $\frac{(s-1)(r+1)^2}{2s}$ [Turan, 1941].

With respect to the growth rate function we are keen on knowing if it is linear or a higher order polynomial like quadratic or exponential. See Nelson’s posts for detailed explanations on this.

In 1963 G. A. Dirac determined the graphs without two vertex disjoint cycles [Dirac, 1963]. Excluding two vertex-disjoint cycles in a 3-connected graph is equivalent to excluding the prism graph $(K_5\backslash e)^*$ as a minor. So essentially Dirac determined the 3-connected graphs with no prism minor. He found that the 3-connected members of this class are $K_5$, $K_5 \backslash e$, the infinite family of wheel graphs $W_r$, for $r\ge 4$, and $K_{3,p}$, $K’_{3,p} $, $K”_{3,p} $ or $K”’_{3,p}$, for some $p\ge 3$.

Theorem 1.  A simple $3$-connected graph has no minor isomorphic to $(K_5\backslash e)^*$ if and only if it is isomorphic to $W_r$ for some $r\ge 3$, $K_5$, $K_5 \backslash e$, $K_{3,p}$, $K’_{3,p} $, $K”_{3,p} $ or $K”’_{3,p}$, for some $p\ge 3$.

Observe that there are two very distinct 3-connected infinite families here: $W_r$ and $K_{3, p}”’$. The size of $W_r$ is $2r+1$ for $r\ge 3$ and the size of $K_{3, r-2}”’$ is $3r-3$ for $r\ge 5$. Both of these 3-connected infinite families are growing linearly. I’ve heard the word “monarchs” used to describe $W_r$ and $K_{3, p}”’$; I think Jack Edmonds used this term. Monarchs is a good word to describe them.

Matroids that are not 3-connected may be pieced together from 3-connected matroids using direct-sums and 2-sums [Oxley, 2012, 8.3.1]. This result was first proven by Bixby in 1972. This is a terrific decomposition result that allows us to focus on just the 3-connected matroids in the family. Thus if we know the 3-connected matroids we can build the 2-connected ones via the operation of two sums.

Now extremal matroid is defined as the simple rank-$r$ matroid of maximum size. So 2-sums and even direct sums are allowed. But when the class contains $W_r$ and $K_{3, r-2}”’$ with $W_r$ growing at rate $2r$ and $K_{3, r-2}”’$ growing at rate $3r-3$, the latter dominates any rank-$r$ graph that can be constructed from direct sum or 2-sum. The proof is straightforward case-checking. Thus the growth rate of the class of graphs with no prism minor is $3r-3$ and the rank-$r$ extremal matroid is $K_{3, r-2}”’$. Even if we were not able to say precisely what the growth rate is, once the 3-connected families are found, and found to be growing linearly, piecing together using direct sum and 2-sum will still give a linear growth rate.

Moving on let’s take a look at Oxley’s 1987 result where he determined the 3-connected matroids with no 4-wheel minor [Oxley, 1987]. The main theorem of that paper is Theorem 2.1 and the key component of the main theorem is Theorem 2.2 which is stated below. Let $Z_r$ denote the $(2r+1)$-element rank-$r$ binary spikes. They are non-regular matroids represented by the binary matrix $[I_r| D]$ where $D$ has $r+1$ columns labeled $b_1, \dots b_r, c_r$. The first $r$ columns in $D$ have zeros along the diagonal and ones elsewhere. The last column is all ones.

Theorem 2. A 3-connected binary non-regular matroid has no minor isomorphic to $P_9$ or ${P_9}^*$ if and only if it is isomorphic to $F_7$, ${F_7}^*$, $Z_r$, ${Z_r}^*$, $Z_r \backslash b_r$, or $Z_r \backslash b_r$, for some $\ge 4$.

Observe that the 3-connected infinite family $Z_r$ is growing at the rate of $2r+1$ elements. In Section 3 of his paper Oxley states without proof (details are straightforward) that the growth rate of the class of binary matroids with no 4-wheel minor is $3r-2$ if $r$ is odd and $3r-3$ if $r$ is even (Theorem 3.1). Why the difference: $2r+1$ versus $3r-2$? The answer is given in the second part of the statement of the theorem, you can 2-sum $F_7$ with itself or $F_7$ with $Z_4$ or $F_7$ with $U_{2,3}$ and get slightly larger rank-$r$ matroids as compared to $Z_r$.

This is why it is sufficient to know the 3-connected members of a class, and for all practical purposes the definition of extremal matroid really should have had 3-connected in it. The definition of extremal matroid made in the 1950s did not forsee Bixby’s result. There’s probably no point changing any definition, but it must be understood clearly that if the 3-connected members of a class are known, then so is the growth rate.

This raises the question: What if we don’t know the 3-connected members, but we have a decomposition result? The first and most well-known such result in Matroid Theory is Paul Seymour’s regular matroid decomposition. He proved that if $M$ is a 3-connected matroid in $EX(F_7, F_7^*)$, then $M$ can be constructed by piecing together graphic and cographic matroids and $R_{10}$, which is a special 10-element matroid [Seymour, 1980].  The  3-connected regular matroids are not known precisely. However, in 1957 Heller showed that the growth rate of the class of regular matroids is $\frac {r(r+1)}{2}$. His proof predates Seymour’s decomposition theorem. I have not seen a proof of growth rate of regular matroids based on the Seymour’s decomposition theorem. Perhaps if someone has, a reference could be mentioned in the comments.

More recently, Kung, Mayhew, Pivotto, and Royle showed the growth rate of $EX(AG(3,2)$ is $\frac {r(r+1)}{2}$. See Nelson’s fifth post on growth-rates. In that post he said:

Given this (and Irene asked a question along these lines in her post), it is natural to ask for a characterisation of exactly which classes of matroids grow like the graphic ones:

In a paper titled “Growth rates of binary matroids with no $P_9^*$-minor,” I found another class that exhibits this behavior.

Theorem 3. Let $M$ be a simple rank-$r$ binary matroid with no $P_9^*$ minor. Then, $|E(M)| \le \frac {r(r+1)}{2}$ with this bound being attained if and only if $M\cong K_{r+1}$. Moreover, the growth rate function of non-regular matroids in $EX[P_9^*]$ is $4r-5$, for $r\ge 6$.

The technique in Oxley’s 1987 paper is Seymour’s Splitter Theorem which was used to obtain the precise 3-connected members. The technique in my paper is the Strong Splitter Theorem, which was joint work with Manoel Lemos, where we built on the Splitter Theorem. Using it I was able to find the 3-connected members for $EX({P_9}^*)$. This approach is completely different from the approach Kung, Mayhew, Pivotto, and Royle adopted to find $EX(AG(3,2)$.

This post is getting quite long. In my next post I will describe the complete structure of the class $EX(P_9^*)$ and explain how I can say precisely the growth rate of the 3-connected non-regular matroids is $4r-5$. Note that $4r-5$ is the growth rate of the rank-$r$ 3-connected family, but it is large enough that it dominates any rank-$r$ 2-connected matroid pieced together from small 3-connected matroids.

The next post will appear on my blog Graphs, Matroids, etc.

References:

[Heller, 1957] I. Heller (1957), On linear systems with integral valued solutions, Pacific J. Math. 7, 1351-1364.

[Kingan, 2015] S. R. Kingan, Growth rates of binary matroids with no $P_9^*$-minor,  arXiv:1412.8169.

[Kingan and Lemos, 2014] S. R. Kingan and M. Lemos (2014), Strong Splitter Theorem Annals of CombinatoricsVol. 18 – 1, 111 – 116.

[Kung, Mayhew, Pivotto, Royle, 2013] J. Kung, D. Mayhew, I. Pivotto, and G. Royle, Maximum size binary matroids with no AG(3,2)-minor are graphic,  http://epubs.siam.org/doi/abs/10.1137/130918915

[Oxley, 1987] J. G. Oxley (1987). The binary matroids with no 4-wheel minor, {\it Trans. Amer. Math. Soc.} {\bf 154}, 63-75.

[Oxley, 2012] J. G. Oxley (2012). {\it Matroid Theory}, Second Edition, Oxford University Press, New York.

[Seymour, 1980] P. D. Seymour (1980) Decomposition of regular matroids, {\it J. Combin. Theory Ser. B} {\bf 28}, (1980) 305-359.

[Turán, 1941] P. Turán (1941), “On an extremal problem in graph theory”, Matematikaiés Fizikai Lapok (in Hungarian) 48: 436–452.

 

4 thoughts on “Extremal Matroids and Growth Rates

  1. Hi Sandra,
    For $r \ge 4$, let $M_r$ be the direct sum of the matroids $M(K_{r-2})$ and $F_7$. Certainly $M_r$ is not regular due to its $F_7$-restriction. Does $M_r$ have a $P_9^*$-minor? (It doesn’t look like it to me, since $P_9^*$ is $3$-connected and nongraphic). If it doesn’t, then for each $r$ the matroid $M_r$ is non-regular with no $P_9^*$-minor, has rank $r$, and has $\binom{r-2}{2} + 7$ elements; this function is much bigger than $4r-5$. What am I missing?

  2. Nothing. You made a good point. I forgot to put 3-connected in the statement of the theorem. (I sent an email to the editor on this to pass along to the referees some months back.) The key component of the growth rate result is finding the 3-connected infinite family Omega_r in the precise matrix form that allows us to conclude its size is 4r-5.

    There should be a term for the largest rank-r 3-connected matroid – I called it extremal – but I can see how that would cause confusion. How about “extremal 3-connected” or “monarch” or “maximal”? Which would be best?

    So I have a question along these lines. If you can’t determine the 3-connected members of excluded minor class as precisely as I did and you just have a decomposition (as in the case of regular matroids and in other classes like EX(E_4)) then how much more work in involved in finding the growth rate? Has anyone written a proof of growth rate of regular matroids using the decomposition of regular matroids?

  3. I don’t know a term for the largest 3-connected matroid. I’m not even fond of ‘maximum-sized’ or ‘extremal’ so I would just tend to use the growth rate function notation for the appropriate subclass of 3-connected matroids.

    I don’t know if anyone has written down a proof for the growth rate for the regular matroids from the decomposition theorem, but it’s certainly easy; one can verify the statement easily for graphic, cographic and $R_{10}$, and then show that you can’t 1-,2-, or 3-sum together two binary matroids with at most $\binom{r+1}{2}$ elements to get a denser matroid, which is a simple calculation.

    More generally, for quadratically (and exponentially) dense classes one can show that the growth rate function is attained by highly connected matroids, so almost any worthwhile structure theorem should imply a growth rate result. For linearly dense classes it might not be needed.

    PS. You linked to my old website; the new one is here: http://www.math.uwaterloo.ca/~apnelson/

  4. Ok then putting the word 3-connected in the last line of Theorem 3 should be enough.

    And I suppose I can add a growth rate theorem to all my decomposition theorems and especially EX(E_4).

    Thanks Peter for your comments.

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