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WHAT IS…

a Matroidal Family ofGraphs?

J. M. S. Simões-PereiraCommunicated by Cesar E. Silva

Figure 1. There are three types of bicycle graphs,including cycles as subgraphs. A graph and its cyclesprovide the simplest example of a matroid. A graphand its bicycles are another example.

In Figure 1, we have three graphs, called bicycles. Theircycles are the triangles and the quadrilateral of the thirdgraph. Given any graph 𝐺, its cycles satisfy the followingtwo conditions:

C1: No cycle contains properly another cycle;C2: If 𝑥 is an edge belonging to two distinct cycles,

then there is a third cycle, in the union of those two, thatdoes not contain 𝑥.

These are the defining axioms of a matroid. In thedefinition of a matroid on a set 𝑆, instead of cycle we usethe word circuit.1

The edge sets of the cycles of a graph𝐺 = (𝑉,𝐸) are thecircuits of a matroid on the edge set 𝐸 of the graph. This

J. M. S. Simões-Pereira is emeritus professor of mathematics at theUniversity of Coimbra (Portugal). His email address is siper@mat.uc.pt.

For permission to reprint this article, please contact:reprint-permission@ams.org.1The “Geometry of Matroids” (page X) gives a dual, equivalent def-inition of a matroid in which the distinguished subsets of 𝑆, calledthe independents, are the ones which do not contain a circuit.

DOI: http://dx.doi.org/10.1090/noti1716

matroid is usually denoted by 𝑃1. If we take as subgraphsthe bicycles instead of the cycles, their edge sets are thecircuits of another matroid on 𝐸, denoted 𝑃2. These factslead us to say that the set of graphs which are cycles,or the set of graphs which are bicycles, form matroidalfamilies of graphs. As a general definition we have:

Definition. A matroidal family of graphs is a nonemptyset 𝑃 of finite, connected graphs such that, given anarbitrary graph 𝐺, the edge sets of subgraphs of 𝐺 whichare members of 𝑃 are the circuits of a matroid on theedge set of 𝐺.

Unfortunately there is no simple generalization to “tri-cycles” or above. Besides the two examples of cycles andbicycles, 𝑃1 and 𝑃2, only one other nontrivial example 𝑃3was known: the circuits are the even cycles together withbicycles that have no even cycles. It was at this point thatwe [3] proved that for any other example of matroidalfamily no two circuits can be homeomorphic. Shortly after-wards, Thomas Andreae and Rüdiger Schmidt discoverednew examples of matroidal families and, indeed, in everyone of their examples, no two circuits are homeomorphic.Consequently, no two graphs are similar in their struc-ture: neither “vertically” (one cannot contain another) nor“horizontally” (one cannot be homeomorphic to another).Nevertheless, in each one of those families, there arealways two graphs, one of them being a minor of theother, as we are assured by the minors theorem, due toNeil Robertson and Paul Seymour, which says that in anyinfinite family of graphs, at least one of them is a minorof another one. Recall that a graph minor is obtainedby deleting edges and vertices and contracting edges.These examples underline the huge difference betweensubgraphs and minors, even though they exhibit similarbehaviors in some areas, such as planarity.

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We are now ready to describe the infinite matroidalfamilies of graphs 𝑃𝑛,𝑟 discovered by Andreae [1]: With 𝑛and 𝑟 integer numbers, 𝑛 ≥ 0 and −2𝑛 + 1 ≤ 𝑟 ≤ 1, thecircuits consist of all edge sets of subgraphs Γ = (𝑉,𝐸)with|𝑉(Γ)| ≥ 2 and |𝐸(Γ)| = 𝑛|𝑉(Γ)|+𝑟, andwhich themselveshave no such subgraphs. For example, for 𝑃1,0 the circuitsare just the cycles and we recover 𝑃1. For 𝑃1,1, the circuitsturn out to be the bicycles and we recover 𝑃2. And for𝑃1,−1 we recover 𝑃0, whose only member is the graph withjust one edge.

The families discovered by Schmidt [2] are even moresurprising. We need the concept of a partly closed setwhich is a set of graphs𝑄 ⊆ 𝑃𝑛,𝑟 and such that, if𝑋,𝑌 ∈ 𝑄,𝑊 = 𝑋∪𝑌, 𝑋 ≠ 𝑊 ≠ 𝑌 and |𝐸(𝑊)| = 𝑛|𝑉(𝑊)| + 𝑟 + 1,then, for each edge 𝑎 ∈ 𝐸(𝑋∩𝑌), there is𝑍 ∈ 𝑄 such that𝑍 is isomorphic to a subgraph of 𝑊−{𝑎}. As an exampleof a partly closed set, we may take the set 𝑄 ⊆ 𝑃1,0 of theeven cycles.

We now define Schmidt’s families: With a partly closedset 𝑄 ⊆ 𝑃𝑛,𝑟, 𝑟 ≤ 0, and 𝑃𝑛,𝑟+1,𝑄 = 𝑄∪𝑃′, where 𝑃′ is theset of all graphs of 𝑃𝑛,𝑟+1 which do not contain a subgraphisomorphic to a graph in 𝑄, we get 𝑃𝑛,𝑟+1,𝑄 as a matroidalfamily. For example, taking 𝑄 to be the set of the evencycles, 𝑃1,1,𝑄 = 𝑄∪𝑃′ where 𝑃′ is the set of bicycles withno even cycle, we recover 𝑃3. Taking𝑄 to be the empty set,which is trivially closed, we recover Andreae’s matroidalfamilies: in fact, 𝑃𝑛,𝑟+1 = 𝑃𝑛,𝑟+1,∅.

Concerning the matroidal families discovered bySchmidt, his most unexpected result is that the numberof such families is uncountably infinite. A question re-mains to challenge our readers: Are there other matroidalfamilies of graphs?

References[1] Thomas Andreae, Matroidal families of finite, connected,

nonhomeomorphic graphs exist, Journal of Graph Theory 2(1978), 149–153. MR493394

[2] Rüdiger Schmidt, On the existence of uncountably manymatroidal families, Discrete Mathematics 27 (1979), 93–97.MR534956

[3] J. M. S. Simões-Pereira, Matroidal Families of Graphs,in Matroid Applications (Neil White, editor), Encyclopediaof Mathematics and its Applications, vol. 40, CambridgeUniversity Press (1992), 91–105. MR1165541

Photo CreditAuthor photo courtesy of J. M. S. Simões-Pereira.

J. M. S. Simões-Pereira

ABOUT THE AUTHOR

J. M. S. Simões-Pereira’s area ofresearch is discrete mathemat-ics. In his spare time, he enjoysplaying piano, grooming his cats,meeting with friends, and readingand speaking languages (besidesPortuguese and English, French,German, Spanish, Italian and a bitof Russian).

948 Notices of the AMS Volume 65, Number 8