nLab
graph minor

Idea

Let $G$ and $H$ be simple graphs. Then $H$ is a graph minor of $G$ if it is isomorphic to a graph obtained by applying a sequence of operations of the following sort, starting from $G$:

• Removing edges;

• Removing isolated points;

• Contracting edges, where contracting an edge means removing it and identifying its endpoints.

Graph minors can be viewed as certain types of subquotients in an appropriate category of graphs.

Categorical POV

We take the point of view that a simple graph is essentially the same thing as a set $V$ equipped with a reflexive symmetric relation $E$, and a morphism of simple graphs is a function which respects these relations. Indeed, such a relation determines and is uniquely determined by a simple graph $G$ where for given vertices $x,y\in V$, there is an edge between $x$ and $y$ in $G$ iff both $E(x,y)$ and $x\ne y$.

Notice that contraction of edges yields a quotient in this category. For example, if we want to contract a collection of edges by identifying certain points along an equivalence relation $R$, that is via some quotient map $q:V\to V/R$, then we simply take the image of the composite

$$E\hookrightarrow V\times V\stackrel{q\times q}{\to }V/R\times V/R;$$E \hookrightarrow V \times V \stackrel{q \times q}{\to} V/R \times V/R;

notice this image is a reflexive symmetric relation on $V/R$. While it might seem more natural to use instead an irreflexive relation to express loop-free graphs (a relation $E$ being irreflexive if it has empty intersection with the diagonal in $V\times V$), the fact of the matter is that irreflexive relations are not closed under the taking of images, and the category of sets with symmetric irreflexive relations and relation-preserving morphisms does not admit the forming of such quotients.

Thus, we let $\mathrm{SimpGph}$ denote the category of sets equipped with a reflexive symmetric relation. This category has many convenient properties:

Theorem

$\mathrm{SimpGph}$ is a Grothendieck quasitopos, in particular regular and even a logos. The opposite category ${\mathrm{SimpGph}}^{\mathrm{op}}$ is also regular.

Proposition

The underlying vertex-set functor $\mathrm{Vert}:\mathrm{SimpGph}\to \mathrm{Set}$ reflects and preserves monos and epis, and preserves all limits and colimits.

For details on these results, see category of simple graphs. A quotient will mean a regular epi $G\twoheadrightarrow H$. A subgraph is a mono $H\hookrightarrow H\prime$ in $\mathrm{SimpGph}$. A subquotient in $\mathrm{SimpGph}$ is a diagram of shape

(1)$$\begin{array}{ccc}G& \twoheadrightarrow & G\prime \\ & & {\uparrow }^{\mathrm{mono}}\\ & & H\end{array}$$\array{ G & \twoheadrightarrow & G' \\ & & \uparrow^\mathrlap{mono} \\ & & H }

Proposition

The subquotient relation is reflexive and transitive.

Proof

Reflexivity is clear. For transitivity, we compose subquotients by taking a pushout square as follows.

$$\begin{array}{ccccc}G& \twoheadrightarrow & G\prime & \twoheadrightarrow & G″\\ & & {}^{\mathrm{mono}}\uparrow & & {\uparrow }^{\mathrm{mono}}\\ & & H& \twoheadrightarrow & H\prime \\ & & & & {\uparrow }^{\mathrm{mono}}\\ & & & & J\end{array}$$\array{ G & \twoheadrightarrow & G' & \twoheadrightarrow & G'' \\ & & ^\mathllap{mono} \uparrow & & \uparrow^\mathrlap{mono} \\ & & H & \twoheadrightarrow & H' \\ & & & & \uparrow^\mathrlap{mono} \\ & & & & J }

where we use the simple fact that the pushout of a mono along an arrow in $\mathrm{SimpGph}$ is a mono (because $\mathrm{Vert}$ reflects monos and preserves monos and pushouts, plus the fact that the pushout of a mono in $\mathrm{Set}$ is a mono).

For finite graphs, the subquotient relation is also antisymmetric (if $G$ and $H$ are minors of one another, then they are isomorphic). Indeed, if either the arrow $G\twoheadrightarrow G\prime$ or the arrow $H\hookrightarrow G\prime$ is not an isomorphism, then $H$ has strictly fewer edges and vertices than $G$. This is clearly not the case for infinite graphs (e.g., the infinite rooted binary tree without leaves contains as a subgraph a disjoint sum of two copies of itself).

(I intend to expand this section, eventually. Hopefully one can develop a categorical story about graph minors in particular.)

Minor-closed families

• The collection of forests is closed under the graph minor relation.

• The collection of planar graphs is closed under the graph minor relation.

Graph minor theorem

This celebrated result of Robertson and Seymour says that no collection of finite graphs, ordered by the graph minor relation, has an infinite antichain. Equivalently, that in any collection of finite graphs, there are only finitely many elements which are minimal with respect to the graph minor relation.

As a corollary, any class $F$ of finite graphs which is closed under minors admits a forbidden minor characterization: is precisely the class of graphs which do not have any members of a specific finite family of graphs as a minor. (Proof: the complement of $F$ in the poset of finite graphs has finitely many minimal members. Then, $x$ belongs to $F$ iff none of these minimal members occurs as a minor of $x$.

For example, the class of forests is precisely the class of finite graphs which do not have a triangle $K_3$ as a minor. The class of planar graphs is precisely the class of finite graphs which do not have $K_5$ or $K_{3,3}$ as a minor.

Forbidden minor characterizations also exist for certain classes of matroids. (See for example Wikipedia here.)

References

• Wikipedia article on graph minors (link)