Studying the structure of graphs is a fundamental topic of broad interest in combinatorial mathematics. At the forefront of this study is the Graph Minor Theorem of Robertson and Seymour , described by Diestel  as “among the deepest theorems mathematics has to offer”. At the heart of the proof of this theorem is the Graph Minor Structure Theorem, which shows that any graph in a minor-closed family222A graph is a minor of a graph if a graph isomorphic to can be obtained from a subgraph of by contracting edges. A class of graphs is minor-closed if for every graph every minor of is in , and some graph is not in . A graph is -minor-free if is not a minor of . can be constructed using four ingredients: graphs on surfaces, vortices, apex vertices, and clique-sums. Graphs of bounded genus, and in particular planar graphs are basic building blocks in graph minor structure theory. Indeed, the theory says nothing about the structure of planar graphs. So it is natural to ask whether planar graphs can be described in terms of some simpler graph classes. In a recent breakthrough, Dujmović et al.  provided an answer to this question by showing that every planar graph is a subgraph of the strong product333The cartesian product of graphs and , denoted by , is the graph with vertex set , where distinct vertices are adjacent if: and ; or and . The strong product of graphs and , denoted by , is the graph with vertex set , where distinct vertices are adjacent if: and ; or and ; or and . If is a subgraph of , then the projection of into is the set of vertices such that for some . of a graph of bounded treewidth444A tree decomposition of a graph is a collection of subsets of (called bags) indexed by the nodes of a tree , such that (i) for every edge , some bag contains both and , and (ii) for every vertex , the set induces a non-empty (connected) subtree of . The width of a tree decomposition is the size of the largest bag minus 1. The treewidth of a graph , denoted by , is the minimum width of a tree decomposition of . See [44, 30, 3, 4, 45] for surveys on treewidth. A path decomposition is a tree decomposition where the underlying tree is a path. The pathwidth of a graph , denoted by , is the minimum width of a path decomposition of . and a path.
Theorem 1 ().
Every planar graph is a subgraph of:
for some graph of treewidth at most and for some path ;
for some graph of treewidth at most and for some path .
This graph product structure theorem is attractive since it describes planar graphs in terms of graphs of bounded treewidth, which are considered much simpler than planar graphs. For example, many NP-complete problem remain NP-complete on planar graphs but are polynomial-time solvable on graphs of bounded treewidth.
Despite being only 10 months old, Theorem 1 is already having significant impact. Indeed, it has been used to solve two major open problems and make additional progress on two other longstanding problems:
Bonamy et al.  use Theorem 1 to give more compact graph encodings of planar graphs. In graph-theoretic terms, this implies the existence of a graph with vertices that contains each planar graph with at most vertices as an induced subgraph, This work improves a sequence of results that goes back 27 years to the introduction of implicit labelling schemes by Kannan et al. .
The first goal of this paper is to introduce several product structure theorems that have been recently established, most of which generalise Theorem 1. First Section 2 considers minor-closed classes. Then Section 3 considers several examples of non-minor-closed classes. Section 4 introduces the notion of graph classes with polynomial growth and their characterisation in terms of strong products of paths due to Krauthgamer and Lee . We prove an extension of this result for strong products of graphs of given pathwidth.
The remaining sections explore how graph product structure might be applicable to more broadly defined graph classes. The following definition by Nešetřil and Ossona de Mendez  provides a setting for this study555Let be the distance between vertices and in a graph . For a vertex in a graph and , let be the set of vertices of at distance exactly from , and let be the set of vertices at distance at most from . The set is called an -ball. We drop the subscript when the graph is clear from the context.. A graph class has bounded expansion with expansion function if, for every graph and for all disjoint subgraphs of radius at most in , every subgraph of the graph obtained from by contracting each into a vertex has average degree at most . When is a constant, is contained in a proper minor-closed class. As is allowed to grow with we obtain larger and larger graph classes. A graph class has linear expansion if has bounded expansion with an expansion function in . A graph class has polynomial expansion if has bounded expansion with an expansion function in , for some constant .
We characterise when a graph class defined by a cartesian or strong product has bounded or polynomial expansion. For and for hereditary666A class of graphs is hereditary if it is closed under induced subgraphs. graph classes and , let
Note that is hereditary. Sections 6 and 5 characterise when has bounded or polynomial expansion. In related work, Wood  characterised when has bounded Hadwiger number, and Pecaninovic  characterised when has bounded Hadwiger number.
Section 7 explores graph product structure theorems for various geometrically defined graph classes. We show that multi-dimensional unit-disc graphs have a product structure theorem, and discusses whether two other naturally defined graph classes might have product structure theorems. We finish with a number of open problems in Section 8.
2 Minor-Closed Classes
Here we survey results generalising Theorem 1 for minor-closed classes. First consider graphs embeddable on a fixed surface777The Euler genus of the orientable surface with handles is . The Euler genus of the non-orientable surface with cross-caps is . The Euler genus of a graph is the minimum Euler genus of a surface in which embeds (with no crossings). See  for background on embeddings of graphs on surfaces..
Theorem 2 ().
Every graph of Euler genus is a subgraph of:
for some graph of treewidth at most and for some path ;
for some graph of treewidth at most and for some path .
for some graph of treewidth at most and some path .
Here is the complete join of graphs and . The proof of Theorem 2 uses an elegant cutting lemma to reduce to the planar case.
Theorem 2 is generalised as follows. A graph is apex if is planar for some vertex .
Theorem 3 ().
For every apex graph , there exists such that every -minor-free graph is a subgraph of for some graph of treewidth at most and some path .
For an arbitrary proper minor-closed class, apex vertices are unavoidable; in this case Dujmović et al.  proved the following product structure theorem.
Theorem 4 ().
For every proper minor-closed class there exist such that every graph can be obtained by clique-sums of graphs such that for ,
for some graph with treewidth at most and some path .
If we assume bounded maximum degree, then apex vertices in the Graph Minor Structure Theorem can be avoided, which leads to the following theorem of Dujmović et al. .
Theorem 5 ().
For every proper minor-closed class , every graph in with maximum degree is a subgraph of for some graph of treewidth and for some path .
It is worth highlighting the similarity of Theorem 5 and the following result of Ding and Oporowski  (refined in ).Theorem 6 says that graphs of bounded treewidth and bounded degree are subgraphs of the product of a tree and a complete graph of bounded size, whereas Theorem 5 says that graphs excluding a minor and with bounded degree are subgraphs of the product of a bounded treewidth graph and a path.
3 Non-Minor Closed Classes
A recent direction pursued by Dujmović et al.  studies graph product structure theorems for various non-minor-closed graph classes. First consider graphs that can be drawn on a surface of bounded genus and with a bounded number of crossings per edge. A graph is -planar if it has a drawing in a surface of Euler genus at most such that each edge is involved in at most crossings. Even in the simplest case, there are -planar graphs that contain arbitrarily large complete graph minors .
Theorem 7 ().
Every -planar graph is a subgraph of , for some graph of treewidth and for some path .
Map and string graphs provide further examples of non-minor-closed classes that have product structure theorems.
Map graphs are defined as follows. Start with a graph embedded in a surface of Euler genus , with each face labelled a ‘nation’ or a ‘lake’, where each vertex of is incident with at most nations. Let be the graph whose vertices are the nations of , where two vertices are adjacent in if the corresponding faces in share a vertex. Then is called a -map graph. A -map graph is called a (plane) -map graph; see [24, 7] for example. The -map graphs are precisely the graphs of Euler genus at most ; see . So -map graphs generalise graphs embedded in a surface, and we now assume that for the remainder of this section.
Theorem 8 ().
Every -map graph is a subgraph of:
, where is a graph with treewidth at most 14 and is a path,
, where is a graph with treewidth and is a path.
A string graph is the intersection graph of a set of curves in the plane with no three curves meeting at a single point; see [41, 25, 26] for example. For , if each curve is in at most intersections with other curves, then the corresponding string graph is called a -string graph. A -string graph is defined analogously for curves on a surface of Euler genus at most .
Theorem 9 ().
Every -string graph is a subgraph of , for some graph of treewidth and some path .
every path in has length at most , and
for every , the number of paths in that use as an internal vertex is at most .
Each path is called a shortcut; if has endpoints and then it is a -shortcut. Given a graph and a -shortcut system for , let denote the supergraph of obtained by adding the edge for each -shortcut in .
Theorem 10 ().
Let be a subgraph of , for some graph of treewidth at most and for some path . Let be a -shortcut system for . Then is a subgraph of for some graph of treewidth at most and some path .
Theorems 9, 8 and 7 are then proved by simply constructing a shortcut system. For example, by adding a dummy vertex at each crossing, Dujmović et al.  noted that every -planar graph is a subgraph of for some graph of Euler genus at most and for some -shortcut system for .
Powers of graphs can also be described by a shortcut system. The -th power of a graph is the graph with vertex set , where if and only if . Dujmović et al.  noted that if a graph has maximum degree , then for some -shortcut system . Theorem 10 then implies:
Theorem 11 ().
For every graph of Euler genus and maximum degree , the -th power is a subgraph of , for some graph of treewidth and some path .
4 Polynomial Growth
This section discusses graph classes with polynomial growth. A graph class has polynomial growth if for some constant , for every graph , for each every -ball in has at most vertices. For example, every -ball in an grid graph is contained in a subgrid, which has size ; therefore the class of grid graphs has polynomial growth. More generally, let be the strong product of infinite two-way paths. That is, where distinct vertices and are adjacent in if and only if for each . Then every -ball in has size at most . Krauthgamer and Lee  characterised the graph classes with polynomial growth as the subgraphs of .
Theorem 12 ().
Let be a graph such that for some constant and for every integer , every -ball in has at most vertices. Then .
We show that a seemingly weaker condition also characterises graph classes with polynomial growth. (We emphasise that in Theorem 13, does not necessarily have bounded maximum degree.)
The following are equivalent for a class of graphs :
has polynomial growth,
there exists such that every graph in is a subgraph of ,
there exist such that for every graph there exist graphs such that:
has maximum degree ,
for each ,
has maximum degree at most for each ,
Krauthgamer and Lee  proved that (1) and (2) are equivalent. It is immediate that (2) implies (3) with and and . So it suffices to show that (3) implies (1). Consider graphs and satisfying (3). For , by Lemma 14 below (with ), every -ball in has at most vertices. By the result of Krauthgamer and Lee , for some . Thus
By Lemma 14 again, every -ball in has size at most
which is at most for some and . Hence (1) holds. ∎
For every graph with pathwidth at most , for every connected subgraph of with radius at most and maximum degree at most ,
The BFS spanning tree of rooted at the centre of has radius at most . So it suffices to prove the result when is a tree. We proceed by induction on with the following hypothesis: For every graph with pathwidth at most , for every subtree of with radius at most and maximum degree at most ,
Since is connected, we may assume that is connected. Since has radius at most ,
where each is a path on vertices.
In the base case , we have and , implying
Now assume that and the claim holds for . Let be the projection of into . Let be a path decomposition of with width . We may delete any bag such that . Now assume that and . Let be a vertex in , and let be a vertex in . Thus and for some . Let be the path in with endpoints and . Since has radius at most , has at most vertices. Let be the set of vertices such that where . Thus . By the choice of and , we have for each . Let . Thus is a path decomposition of with width at most . Let . Thus . Let . Hence is a subgraph of . Each component of has a neighbour in , implying that has at most components. Every subtree of has radius at most (centred at the vertex closest to the centre of ). By induction, each component of has at most vertices. Thus
as desired. ∎
Property (3) in Theorem 13 is best possible in a number of respects. First, note that we cannot allow and to have unbounded maximum degree. For example, if and are both , then and both have pathwidth 1, but contains as a subgraph, which contains a complete binary tree of height, which is a bounded-degree graph with exponential growth. Also, bounded pathwidth cannot be replaced by bounded treewidth, again because of the complete binary tree.
5 Polynomial Expansion
This section characterises when has polynomial expansion. Separators are a key tool here. A separation in a graph is a pair of subgraphs of such that and . The order of is . A separation is balanced if and . A graph class admits strongly sublinear separators if there exists and such that for every graph , every subgraph of has a balanced separation of order at most . Dvořák and Norin  noted that a result of Plotkin et al.  implies that graph classes with polynomial expansion admit strongly sublinear separators. Dvořák and Norin  proved the converse (see [16, 20, 23] for more results on this theme).
Theorem 15 ().
A hereditary class of graphs admits strongly sublinear separators if and only if it has polynomial expansion.
Robertson and Seymour  established the following connection between treewidth and balanced separations.
Lemma 16 ([47, (2.6)]).
Every graph has a balanced separation of order at most .
Dvořák and Norin  proved the following converse.
Lemma 17 ().
If every subgraph of a graph has a balanced separation of order at most , then .
We have the following strongly sublinear bound on the treewidth of graph products.
Let be an -vertex subgraph of for some graph . Then
Let . For , let be the layering of determined by the -th dimension. Let
For and , let . Thus is a partition of . Hence for some . Let . Thus . Note that each component of is a subgraph of , where is the path on vertices. Since equals the maximum treewidth of the connected components of , we have . To obtain a tree decomposition of with width , start with an optimal tree decomposition of , and replace each instance of a vertex of by the corresponding copy of . Thus
Lemma 18 is generalised by our next result, which characterises when a graph product has polynomial expansion. The following definition is key. Say that graph classes and have joint polynomial growth if there exists a polynomial function such that for every , there exists such that for every graph every -ball in has size at most .
The following are equivalent for hereditary graph classes and :
has polynomial expansion,
has polynomial expansion,
has polynomial expansion, has polynomial expansion, and and have joint polynomial growth.
(1) implies (2) since .
We now show that (2) implies (3). Assume that has polynomial expansion. That is, for some polynomial , for every graph , every -shallow minor of has average degree at most . Since , both and have polynomial expansion.
Assume for the sake of contradiction that and do not have joint polynomial growth. Thus for every polynomial there exists such that for each some -ball of some graph has at least vertices. Apply this where is a polynomial with , where . Since and are hereditary, there exists such that there is a graph with radius at most and at least vertices, and there is a graph with radius at most and at least vertices.
Let be the central vertex in . Since , for some , there is a set of vertices in at distance exactly from . For all , let be the shortest -path contained with the union of a shortest -path and a shortest -path in . Thus has length at most and . Let be a set of vertices in . Fix an arbitrary bijection .
Let . For each , let ; note that is isomorphic to , and thus has radius at most . Moreover, and are disjoint for distinct . For , let ; note that is isomorphic to . Let be the copy of the path within . Since , the only vertices of in are and , which are the endpoints of in and respectively. Since has length at most , so does .
By construction, and are disjoint for distinct . Contract to a vertex for each , and contract to an edge for each . We obtain the complete graph as a minor in . Moreover, the minor is -shallow. This is a contradiction, since has average degree greater than .
We prove that (3) implies (1) by a series of lemmas below (culminating in Lemma 23 below). ∎
For a graph , a set is -localising if for every component of , there exists a vertex such that for every (note that the distance is in , not in ).
The following is a variation on Lemma 5.2 of Krauthgamer and Lee . For and with , consider the following function defined on . First, let . Now, for every integer , inductively define
Fix and with , such that (so defines a probability distribution on
defines a probability distribution on). For every graph , there exists a probability distribution over the -localising subsets of such that the set drawn from this distribution satisfies for every .
Let . For , choose independently at random such that . For each , let be the minimum index such that , and let .
First we argue that is -localising. Consider any component of , and let be the vertex of with minimum. Suppose for the sake of contradiction that contains a vertex at distance at least from , and let be a path from to in . Then contains a vertex at distance exactly from . However, since , we conclude and , which is a contradiction.
Next, we bound the probability that a vertex of is in . Consider any . If , then . If , then . If , then letting we have
For every polynomial , there exists such that the following holds. Let be a positive integer and let be a graph such that for every . Then there exists a probability distribution over the -localising subsets of such that the set drawn from this distribution satisfies for every .
Let be the degree of plus one, so that for every sufficiently large . Let and . Note that for sufficiently large ,
Hence . It follows by induction that for each . Thus and . The claim follows from Lemma 20. ∎
For every polynomial , there exists such that the following holds. Let be a positive integer and let be a graph such that for every . Then for every function , there exists such that and each component of has at most vertices.
Choose an -localising set using Corollary 21. Since is -localising and for every , each component of has at most vertices. Furthermore,
Hence there is a choice for such that . ∎
Suppose and are classes with strongly sublinear separators and of joint polynomial growth (bounded by a polynomial ). Then has strongly sublinear separators.
Let be such that every subgraph of a graph from has a balanced separator of order at most . Let be sufficiently small (depending on and ).
Suppose , , and is a subgraph of . Let and be the projections from to and . Let and . By symmetry, we may assume for every vertex of . Let for each . By Corollary 22, there exists such that and each component of has at most vertices. Let ; then .