Sparsity and dimension
We prove that posets of bounded height whose cover graphs belong to a fixed class with bounded expansion have bounded dimension. Bounded expansion, introduced by Nešetřil and Ossona de Mendez as a model for sparsity in graphs, is a property that is naturally satisfied by a wide range of graph classes, from graph structure theory (graphs excluding a minor or a topological minor) to graph drawing (e.g. graphs with bounded book thickness). Therefore, our theorem generalizes a number of results including the most recent one for posets of bounded height with cover graphs excluding a fixed graph as a topological minor. We also show that the result is in a sense best possible, as it does not extend to nowhere dense classes; in fact, it already fails for cover graphs with locally bounded treewidth.
Key words and phrases:Bounded expansion, poset, dimension, cover graph, graph minor
1.1. Poset dimension and cover graphs
Partially ordered sets, posets for short, are studied extensively in combinatorics, set theory and theoretical computer science. One of the most important measures of complexity of a poset is its dimension. The dimension of a poset is the least integer such that points of can be embedded into in such a way that in if and only if the point of is below the point of with respect to the product order on . Equivalently, the dimension of is the least such that there are linear extensions of whose intersection is . Not surprisingly, dimension is hard to compute: Already deciding whether is an NP-hard problem , and for every , there is no -approximation algorithm for dimension unless ZPP=NP, where denotes the size of the input poset . Research in this area typically focus on finding witnesses for large dimension and sufficient conditions for small dimension; see e.g.  for a survey.
Posets are visualized by their Hasse diagrams: Points are placed in the plane and whenever in the poset, and there is no point with , there is a curve from to going upwards (that is -monotone). The diagram represents those relations which are essential in the sense that they are not implied by transitivity, known as cover relations. The undirected graph implicitly defined by such a diagram is the cover graph of the poset. That graph can be thought of as encoding the ‘topology’ of the poset.
There is a common belief that posets having a nice or well-structured drawing should have small dimension. But first let us note a negative observation by Kelly , from 1981, there is a family of posets whose diagrams can be drawn without edge crossings—undoubtedly qualifying as a ‘nice’ drawing—and with arbitrarily large dimension, see Figure 1. A key observation about Kelly’s construction is that these posets also have large height. This leads us to our main theorem, which generalizes several previous works in this area.
For every class of graphs with bounded expansion, and for every integer , posets of height whose cover graphs are in have bounded dimension.
Theorem 1 takes its roots in the following result of Streib and Trotter : Posets with planar cover graphs have dimension bounded in terms of their height. Note that this justifies the fact that height and dimension grow together in Kelly’s construction. Joret, Micek, Milans, Trotter, Walczak, and Wang  subsequently proved that the same result holds in the case of cover graphs of bounded treewidth, of bounded genus, and more generally cover graphs that forbid a fixed apex graph as a minor. (Recall that a graph is apex if it can be made planar by removing at most one vertex.) In another direction, Füredi and Kahn  showed that posets with cover graphs of bounded maximum degree have dimension bounded in terms of their height.111We note that the original statement of Füredi and Kahn’s theorem is that posets with comparability graphs of bounded maximum degree have bounded dimension; in fact, they show a upper bound on the dimension, where denotes the maximum degree. Observe however that the comparability graph of a poset has bounded maximum degree if and only its cover graph does and its height is bounded. All this was recently generalized by Walczak : Posets whose cover graphs exclude a fixed graph as a topological minor have dimension bounded by a function of their height. (We note that Walczak’s original proof relies on the graph structure theorems for graphs excluding a fixed topological minor; see  for an elementary proof.)
Bounded degree graphs, planar graphs, bounded treewidth graphs, and more generally graphs avoiding a fixed (topological) minor are sparse, in the sense that they have linearly many edges. This remark naturally leads one to ponder whether simply having a sparse cover graph is enough to guarantee the dimension be bounded by a function of the height. This is exactly the question we address in this work. Our main contribution is to characterize precisely the type of sparsity that is needed to ensure the property. Before explaining it let us make some preliminary observations.
Clearly, asking literally that the cover graph has at most edges for some constant , where is the number of points of the poset, is not enough since one could simply consider the union of a standard example on points and a large antichain. Even requiring that property to hold for every subgraph of the cover graph (that is, bounding its degeneracy) is still not enough: Define the incidence poset of a graph as the height- poset with point set , where for a vertex and an edge we have in whenever is an endpoint of in (and no other relation). The fact that follows easily from repeated applications of the Erdős-Szekeres theorem for monotone sequences (see for instance ). Yet, the cover graph of is a clique where each edge is subdivided once, which is -degenerate.
1.3. Nowhere dense classes and classes with bounded expansion
Nešetřil and Ossona de Mendez  carried out a thorough study of sparse classes of graphs over the last decade. Most notably, they introduced the notions of nowhere dense classes and classes with bounded expansion. Let us give some informal intuition, see Section 2 for the precise definitions. The key idea in both cases is to look at minors of bounded depth, that is, minors that can be obtained by first contracting disjoint connected subgraphs of bounded radius, and then possibly removing some vertices and edges. In a nowhere dense class it is required that bounded-depth minors exclude at least some graph (which can depend on the depth). In a class with bounded expansion the requirement is stronger: For every , depth- minors should be sparse, that is, their average degrees should be bounded by some function .
It is well-known that graphs with no -minors have average degree bounded by a function of . Thus the corresponding class of graphs has bounded expansion, since the average degree of their minors is uniformly bounded by a constant. The class of graphs with no topological -minors also has bounded expansion, though in that case the bounding function might not be constant anymore, see . These remarks show that Theorem 1 generalizes the aforementioned result for cover graphs excluding a fixed graph as a topological minor, and hence the previous body of work as well.
In Figure 2 we give a summary of all the known results about posets with sparse cover graphs having their dimension bounded by a function of their height. We also mention the cases where dimension is bounded by an absolute constant, see the bottom part of the figure. Complementing our main result, we describe in Section 2 a family of height- posets whose cover graphs form a nowhere dense class and having unbounded dimension. Thus Theorem 1 cannot be extended to nowhere dense classes. The graphs constructed moreover have locally bounded treewidth, showing that this property alone is not sufficient either. Finally, in order to complete our study of sparse cover graphs following the theory of Nešetřil and Ossona de Mendez, we remark that another specialization of nowhere dense classes called almost wide classes also fail, in the sense that dimension is not bounded in terms of height for posets with cover graphs belonging to such a class. (We postpone the rather technical definition of almost wide classes until Section 2.)
Let us now turn our attention to some applications of Theorem 1 in the context of graph drawing, where natural classes with bounded expansion appear that do not fit in the previous setting of excluding a (topological) minor. Consider for instance posets whose diagrams can be drawn with ‘few’ edge crossings. If we bound the total number of crossings, then the cover graphs have bounded genus, and hence earlier results apply. On the other hand, if we only bound the number of crossings per edge in the drawing, say at most such crossings, then we come to the well-studied class of -planar graphs. While -planar graphs are sparse—Pach and Tóth  proved that their average degree is at most — they do not exclude any graph as a topological minor (assuming ), since every graph has a -planar subdivision: just start with any drawing of the graph in the plane and subdivide its edges around each crossing. Note however that this could require some edges to be subdivided many times. Indeed, one can observe that if an -subdivision of a graph is -planar then is -planar. Combining this fact with Pach and Tóth’s result, Nešetřil, Ossona de Mendez, and Wood  proved that the class of -planar graphs has bounded expansion. Therefore, by Theorem 1, whenever a poset has a -planar cover graph, is bounded by a function of and the height of .
Another example is given by graphs with bounded book thickness. A book embedding of a graph is a collection of half-planes (pages) all having the same line as their boundary (the spine) such that all vertices of the graph lie on the line, every edge is contained in one of the half-planes, and edges on a same page do not cross. The book thickness (also known as stack number) of a graph is the smallest number of pages in a book embedding. Clearly, every graph on vertices with book thickness has at most edges. Still, every graph has a subdivision with book thickness at most (see e.g. ). Nešetřil et al.  observed that a result of Enomoto, Miyauchi, and Ota  easily implies that every class of graphs with bounded book thickness has bounded expansion. Therefore, by Theorem 1, whenever a poset has a cover graph of book thickness at most , its dimension is bounded by a function of and the height of .
Yet another class of graphs with bounded expansion from graph drawing is that of graphs with bounded queue number. A queue layout of a graph is an ordering of its vertices (the spine) together with an edge coloring such that there are no two nested monochromatic edges, where two edges are nested if all four endpoints are distinct and the endpoints of one edge induce an interval on the spine containing the endpoints of the other edge. Then the queue number of a graph is the minimum number of colors in a queue layout. Every graph has a subdivision with queue number , as proved by Dujmović and Wood , who also showed that graphs with bounded queue number form a class with bounded expansion. A challenging open problem in the area of queue layouts is to decide whether planar graphs have constant queue number (see e.g. ).
1.5. Proof overview
In the proof we use a characterization of classes with bounded expansion in terms of -centered colorings. A -centered coloring of a graph is a vertex coloring of such that, for every connected subgraph of , either some color is used exactly once in , or at least colors are used in . Nešetřil and Ossona de Mendez  proved that a class has bounded expansion if and only if there is a function such that for every integer and every graph , there is a -centered coloring of using at most colors. Given a poset of height with cover graph in , we work with a -centered coloring of the cover graph with at most colors.
Given this coloring , we focus on upsets of points : For each point such that in , we consider a sequence of cover relations witnessing that , which we call a covering chain (thus this is a path from going up to in the diagram of ). Such a covering chain defines a corresponding color sequence , whose length is bounded by the height of . We then exploit the fact that there are a bounded number of different color sequences and consider upsets of points w.r.t. a fixed color sequence . Focusing on such upsets, we uncover a wealth of structure, culminating in a proof that a certain family of subsets of points of is laminar. In fact, we define one such laminar family for each subset of the set of color sequences. Once this is set up, we then exploit the underlying tree structure of these laminar families to bound the dimension of .
We note that the idea for the first part of the proof—working with colored-upsets and proving that certain well-chosen families of sets are laminar—comes directly from a recent paper of Reidl, Sánchez Villaamil, and Stavropoulos , who used it to obtain a new characterization of classes with bounded expansion in terms of ‘neighborhood complexity’.
Indeed, their proof method turned out to be perfectly fitted to approach our problem in a clean and simple way.
As a result, the proof we present in this paper is shorter and simpler than the one we gave in the preliminary version of this paper .
Without going into details, that proof relied on a powerful decomposition of posets into layers called unfolding, introduced in the paper of Streib and Trotter , that emerged as a key technical tool in recent papers on poset dimension.
We see it as an appealing feature of our proof that it avoids unfolding entirely.
In fact, no background nor tool from poset theory is needed, except for the elementary notion of an alternating cycle (see Section 2.1).
The paper is organized as follows. In Section 2 we give the necessary definitions for all sparse classes of graphs depicted in Figure 2, as well as the necessary definitions regarding posets. Along the way, we also describe constructions of posets showing that Theorem 1 cannot be extended to the classes in grey in Figure 2. In Section 3 we give the proof of Theorem 1.
2. Definitions and preliminaries
Let us start with basic notions about posets. All posets considered in this paper are finite. Elements of a poset are called points. Points are said to be comparable in if or in . Otherwise and are incomparable in . A set of points is a chain in if the points in are pairwise comparable. The height of is the maximum size of a chain in . We write in if it holds that and . For distinct , if in and there is no with in then is a cover relation of . By we denote the cover graph of , the (undirected) graph defined on the points of where edges correspond to cover relations of .
A linear extension of is a linear order on the ground set of such that in whenever in . Linear extensions form a realizer of if their intersection is equal to , that is, in if and only if in for each . The dimension of , denoted by , is the least number such that there is a realizer of of size .
We let denote the set of ordered pairs of incomparable points in . A set of incomparable pairs is reversible if there is a linear extension of that reverses each pair in , that is, in for every .
We can rephrase the definition of dimension as follows: Assuming , the dimension of is the least positive integer for which there exists a partition of into reversible sets. (Note that in case .) In light of this statement, it is handy to have a simple criterion to decide when a set of incomparable pairs is reversible. This motivates the following definition. A sequence of pairs from such that holds in for all (cyclically) is called an alternating cycle. It is well known (and easy to show) that a set of incomparable pairs is reversible if and only if does not contain any alternating cycle (see e.g. ). Indeed, this is the only fact about dimension that we will need in our proof.
2.2. Sparse graph classes
Next, we introduce the necessary definitions regarding graphs and give proper definitions for the graph classes mentioned in the introduction. All graphs in this paper are finite, simple, and undirected. Given a graph we denote by and the vertex set and edge set of , respectively. is a subgraph of if and . For a subset we denote by the subgraph of with vertex and all edges with both endpoints in . The distance between two vertices in is the length of a shortest path between them. (Thus adjacent vertices are at distance ; also, distance between two vertices in distinct components of is set to .) The set of all vertices at distance at most from vertex in is denoted by , and the subscript is omitted if is clear from the context. The radius of a connected graph is the least integer for which there is a vertex such that .
The treewidth of a graph , denoted by , is the least integer for which there is a tree and a family of non-empty subtrees of such that for each node of , and for each edge . A class of graphs has locally bounded treewidth if there exists a function such that for every integer , graph and vertex .
Given a partition of the vertices of a graph into non-empty parts inducing connected subgraphs, we denote by the graph with vertex set and edge set defined as follows: For two distinct distinct parts , we have if and only if there exist and such that . A graph is a minor of if is isomorphic to a subgraph of for some such partition of . A specialization of this notion is that of topological minors: is a topological minor of a graph if contains a subgraph isomorphic to a subdivision of . (A subdivision of is any graph that can be obtained from by replacing each with a path having and as endpoints and whose internal vertices are new vertices of the graph, that is, the paths () are internally vertex-disjoint.) A class of graphs is minor closed (topologically closed) if every minor (topological minor, respectively) of a graph in is also in .
We pursue with the definitions of classes with bounded expansion and nowhere dense classes. A graph is a depth- minor (also known as an -shallow minor) of a graph if is isomorphic to a subgraph of for some partition of into non-empty parts inducing subgraphs of radius at most . The greatest reduced average density (grad) of rank of a graph , denoted by , is defined as . A class of graphs has bounded expansion if there exists a function such that for every integer and graph . A class of graphs is nowhere dense if for each integer there exists a graph which is not a depth- minor of any graph .
It is easy to see that classes with locally bounded treewidth and classes with bounded expansion are nowhere dense. Note however that these two notions are incomparable. Two classical examples of classes with locally bounded treewidth are graphs with bounded maximum degree, and graphs excluding some apex graph as a minor. The latter is in fact a characterization of minor-closed classes with locally bounded treewidth: A minor-closed class has locally bounded treewidth if and only if excludes some apex graph, a fact originally proved by Eppstein .
Let us now define almost wide classes. For , a set of vertices in a graph is -independent if every two distinct vertices in are at distance strictly greater than in . A class of graphs is almost wide if there exists an integer such that for every integer there is a function such that for every integer , every graph of order at least contains a subset of size at most so that has a -independent set of size . Nešetřil and Ossona de Mendez [19, Theorem 3.23] proved that a class of graphs excluding a fixed graph as a topological minor is almost wide. Moreover, they proved [19, Theorem 3.13] that every hereditary class of graphs that is almost wide is also nowhere dense. (Recall that a class is hereditary if it is closed under taking induced subgraphs.)
2.3. Posets with cover graphs in a nowhere dense class
As mentioned in the introduction, the statement of Theorem 1 cannot be pushed further towards nowhere dense classes. In fact, it already fails for classes with locally bounded treewidth and hereditary classes that are almost wide, as we now show. Our construction is based on the class of graphs with , where and denote the maximum degree and girth of , respectively. This is a useful example of a hereditary class with locally bounded treewidth that is also almost wide but that does not have bounded expansion. (Indeed, this class is invoked several times in the textbook .) We will use in particular that the chromatic number of these graphs is unbounded. This is a well-known fact that can be shown in multiple ways; we can note for instance that the chromatic number of the -vertex -regular non-bipartite Ramanujan graphs with girth built by Lubotzky, Phillips, and Sarnak  have chromatic number (see ).
There exists a hereditary almost wide class of graphs with locally bounded treewidth such that posets of height with cover graphs in have unbounded dimension.
For a graph , the adjacency poset of is the poset with point set such that, for every two distinct vertices , we have in if and only if . It is well known that , see . This can be seen as follows. Fix a realizer of . For every vertex , fix a number such that in . We claim that is a proper coloring of . Consider any two adjacent vertices and in and, say, . Then in , which witnesses that . Therefore, .
Now let denote the class of graphs satisfying . As mentioned earlier, this class is hereditary, almost wide, and has locally bounded treewidth. This is not difficult to check (or see  for a proof).
The key observation about the class in this context is that if , then . This can be seen as follows: First, clearly , so it is enough to show . To show the latter, we remark that if is a cycle of , then naturally corresponds to a closed walk in of the same length. Moreover, every three consecutive vertices in that walk are pairwise distinct, as follows from the adjacency poset construction. Hence, contains a cycle, which is of length at most that of . Therefore, , as claimed.
To summarize, graphs in have unbounded chromatic number, implying that adjacency posets of these graphs have unbounded dimension. Yet, the cover graphs of these adjacency posets all belong to , a hereditary almost wide class with locally bounded treewidth. This concludes the proof. ∎
3. Proof of main theorem
There are a number of equivalent conditions for classes of graphs to have bounded expansion (see  for instance). As mentioned in the introduction, we use the following characterization in terms of -centered colorings: A class has bounded expansion if and only if there exists a function such that for every integer and every graph there is a -centered coloring of using at most colors. Thus, the following theorem implies Theorem 1.
If is a poset of height and its cover graph has a -centered coloring using colors, then
This section is devoted to the proof of Theorem 3. Let thus be a poset of height , and let be a -centered coloring of the cover graph of using colors from the set .
3.1. Signatures for covering chains
First, we refine the coloring of the cover graph of as follows. For each , let denote the height of in , that is the size of a longest chain in ending with . For each we define
Note that is a -centered coloring as in general any refinement of a -centered coloring is still a -centered coloring, for every .
We say that is a covering chain of if is a cover relation in for each . In this case, we also say that is a covering chain from to . Each covering chain has a signature defined by
When is the signature of , we also call a -covering chain. By our definition of we have that the signature of is always proper, in the sense that no color appears more than once in the signature.
Let be the set of all signatures of covering chains of . We can bound the size of as follows: . For each , we let denote the set of all points in such that there is a -covering chain starting from . For illustration, each in belongs to for . For in , the -upset of is the set of all points in such that there is a -covering chain from to . Similarly, the -downset of is the set of all points in such that there is a -covering chain from to .
3.2. Two lemmas on -upsets
Let and let . If a -covering chain starting in intersects a -covering chain starting in , then . In particular, if , then .
Let , be two -covering chains from to and from to , respectively. Suppose that these two paths share a common point . We need to prove that . Suppose for contradiction that , and assume without loss of generality that . Let . Let be a -covering chain from to (see Figure 3 for an illustration).
Clearly, the union of the three paths , and is a connected subgraph of the cover graph of . Since these three paths have the same signature, we know that is colored by with at most colors.
We claim that the two paths and are vertex disjoint. Suppose that and share a common element . Combining the – portion of with the – portion of we obtain a covering chain from to . Using that the signature is proper, it is easy to see that this covering chain from to also has signature . It follows that , contradicting our choice of . Hence, and are vertex disjoint, as claimed. It follows that no color of is used exactly once on , contradicting the fact that is a -centered coloring. ∎
For and points , write if and intersect. By Lemma 4 we know that defines an equivalence relation on ; denote by the equivalence class of point with respect to .
For , let denote the set of points of for which is exactly the subset of signatures such that there is a -covering chain starting at , that is,
Let be the family of subsets of defined as the union of the equivalence classes of for restricted to the set . That is,
A key observation is that these families are laminar. Recall that a family is laminar if for every we have , or , or .
The family is laminar for every .
Let and let . We need to prove that , or , or . Let and be such that and . In order to get a contradiction, suppose that , , and . Thus, there are elements such that , , and . It follows that and . Since and we know that the -upsets and -upsets of , and are not empty. Hence, there exist , , , and . Let denote a -covering chain from to , and define , , , , similarly.
Consider the union of all these paths:
(See Figure 3 for an illustration). Clearly, is a connected subgraph of the cover graph of , and all colors of used on come from and . Both and consist of at most colors and we know moreover that they share at least one color, namely . This means that is colored by with at most colors.
Next, we show that each color of appears at least twice on . To do so, we show that the two paths and are vertex disjoint. Indeed, if these two paths intersect then by Lemma 4 we would have and therefore , contradicting the choice of .
Analogously, we argue that each color of appears at least twice on by showing that the two paths and are vertex disjoint. Indeed, if these two paths intersect then by Lemma 4 we would have and therefore , contradicting the choice of .
This shows that among the at most colors used by on , none appears exactly once, contradicting the fact that is a -centered coloring. ∎
Using the family , we define a linear order on . This linear order results from the following recursive procedure applied to : If no set in is a proper subset of , then order the elements in arbitrarily. Otherwise, let denote the inclusion-wise maximal sets in that are proper subsets of . Since is laminar no two of the sets intersect. Moreover, the union of is . (Indeed, for some and . Thus, for each , the set intersects and avoids , which implies and , by laminarity.) We set for all and all and . The relative ordering of the elements within () is then obtained by applying the recursive procedure to .
For distinct elements we say that lies to the left of in if , and to the right of in otherwise. A crucial property of is that for every and the points of form an interval in . We emphasize this property with the following proposition.
Let , and . If and , then .
3.3. Partitioning the incomparable pairs
Using the tools developed so far, we now show how to partition the set of incomparable pairs of into a number of reversible sets that depends only on . For let . Clearly,
Since , there are at most sets in the above partition of . Note that this partition of is based solely on properties of the first element of pairs in . We are going to refine this partition by considering properties of the second element of pairs in . We need the following observation:
Let , , and . Then no two points are such that .
Suppose to the contrary that there are points such that . Since , we have . But by Proposition 6 this yields , forcing in , a contradiction. ∎
For each and we let be the binary vector where
For , let . Clearly,
We show that each set in this partition of is reversible.
is reversible for every and .
Let and , and suppose for a contradiction that is not reversible. Then contains an alternating cycle . Shifting cyclically the pairs if necessary, we may assume that is leftmost among all ’s with respect to . Consider a covering chain from to and another from to (if then ). Say they have signatures and respectively, and denote them by and .
Clearly, . Since and , we conclude that . Similarly since and , we have by Lemma 7. As we also obtain that and . (In particular, this shows .)
Since we know that there is with such that there is a -covering chain from to . Since and , there is also a -covering chain from to , for some in . And finally, since and , there is a -covering chain from to , for some in . See Figure 4 for a possible situation. Now we consider the union
is a connected subgraph of the cover graph of , and its vertives are colored by with colors coming from and . Since and share at least one common color, namely , we see that at most different colors appear on .
We claim that the two paths and are vertex disjoint. Suppose not. Then by Lemma 4 . Since , by Proposition 6 this implies that . However, this implies in turn that , and in particular , contradicting the fact that and are incomparable in .
Next we claim that and are also vertex disjoint. Arguing again by contradiction, suppose it is not the case. Then by Lemma 4. Since , by Proposition 6 it follows that , which implies , contradicting the fact that and are incomparable.
We conclude that every color from is used at least twice on (since and are disjoint), and that the same holds for colors from (since and are disjoint). This contradicts the fact that is a -centered coloring of the cover graph of . ∎
Since we have at most ways to choose . Also, for we have . Therefore, we can bound the dimension of as follows:
This concludes the proof of Theorem 3.
4. Concluding Remarks
Our main theorem reveals a connection between the notions of bounded expansion and order dimension. We think that this might actually provide a new characterization of classes with bounded expansion:
Let be a class of graphs closed under taking subgraphs. Then has bounded expansion if and only if for each , posets of height whose cover graphs are in have bounded dimension.
Note that Theorem 1 shows one of the two directions of this conjecture, the other one remains open.
As mentioned earlier, classes with bounded expansion can be characterized in various ways by asking that certain coloring numbers are bounded (this includes e.g. low tree-depth colorings, generalised colorings, and centered colorings, see ). In most cases, if one asks instead the coloring numbers to be subpolynomially bounded in terms of the number of vertices of the graph then this results in a characterization of nowhere dense classes. For instance, a class is nowhere dense if and only if for every and , every -vertex graph in has a -centered coloring with colors; see  for a survey of such characterizations. Perhaps one could similarly establish a subpolynomial bound on the dimension of bounded-height posets with cover graphs in a nowhere dense class? (This was suggested by Dan Král.)
Is it true that if is a nowhere dense class of graphs, then for every and , posets on points of height and with cover graphs in have dimension ?
We thank David R. Wood for suggesting that Theorem 1 might be true, which prompted us to work on this question, and Bartosz Walczak for his insightful remarks on the proof. We are also grateful to the referees of the current version of this paper and of its preliminary version  for their numerous and helpful comments.
-  Csaba Biró, Mitchel T. Keller, and Stephen J. Young. Posets with cover graph of pathwidth two have bounded dimension. Order, 33(2):195–212, 2016. arXiv:1308.4877.
-  Parinya Chalermsook, Bundit Laekhanukit, and Danupon Nanongkai. Graph products revisited: tight approximation hardness of induced matching, poset dimension and more. In Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1557–1576. SIAM, Philadelphia, PA, 2012.
-  Vida Dujmović. Graph layouts via layered separators. J. Combin. Theory Ser. B, 110:79–89, 2015. arXiv:1302.0304.
-  Vida Dujmović and David R. Wood. Stacks, queues and tracks: layouts of graph subdivisions. Discrete Math. Theor. Comput. Sci., 7(1):155–201, 2005.
-  Ben Dushnik and Edwin W. Miller. Partially ordered sets. Amer. J. Math., 63:600–610, 1941.
-  Hikoe Enomoto, Miki Shimabara Miyauchi, and Katsuhiro Ota. Lower bounds for the number of edge-crossings over the spine in a topological book embedding of a graph. Discrete Appl. Math., 92(2-3):149–155, 1999.
-  David Eppstein. Diameter and treewidth in minor-closed graph families. Algorithmica, 27(3-4):275–291, 2000. arXiv:math/9907126.
-  Stefan Felsner and William T. Trotter. Dimension, graph and hypergraph coloring. Order, 17(2):167–177, 2000.
-  Stefan Felsner, William T. Trotter, and Veit Wiechert. The Dimension of Posets with Planar Cover Graphs. Graphs Combin., 31(4):927–939, 2015.
-  Zoltan Füredi and Jeff Kahn. On the dimensions of ordered sets of bounded degree. Order, 3(1):15–20, 1986.
-  Martin Grohe, Stephan Kreutzer, and Sebastian Siebertz. Characterisations of nowhere dense graphs. In 33nd International Conference on Foundations of Software Technology and Theoretical Computer Science, volume 24 of LIPIcs. Leibniz Int. Proc. Inform., pages 21–40. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2013.
-  Gwenaël Joret, Piotr Micek, Kevin G. Milans, William T. Trotter, Bartosz Walczak, and Ruidong Wang. Tree-width and dimension. Combinatorica, 36(4):431–450, 2016. arXiv:1301.5271.
-  Gwenaël Joret, Piotr Micek, William T. Trotter, Ruidong Wang, and Veit Wiechert. On the dimension of posets with cover graphs of treewidth . Order, to appear. arXiv:1406.3397.
-  Gwenaël Joret, Piotr Micek, and Veit Wiechert. Sparsity and dimension. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ’16, pages 1804–1813. SIAM, 2016. arXiv:1507.01120v1.
-  David Kelly. On the dimension of partially ordered sets. Discrete Math., 35:135–156, 1981.
-  Alexander Lubotzky, Ralph Phillips, and Peter Sarnak. Ramanujan graphs. Combinatorica, 8(3):261–277, 1988.
-  Piotr Micek and Veit Wiechert. Topological minors of cover graphs and dimension. Submitted, arXiv:1504.07388.
-  Jaroslav Nešetřil and Patrice Ossona de Mendez. Grad and classes with bounded expansion. I. Decompositions. European J. Combin., 29(3):760–776, 2008. arXiv:math/0508323.
-  Jaroslav Nešetřil and Patrice Ossona de Mendez. First order properties on nowhere dense structures. J. Symbolic Logic, 75(3):868–887, 2010.
-  Jaroslav Nešetřil and Patrice Ossona de Mendez. Sparsity, volume 28 of Algorithms and Combinatorics. Springer, Heidelberg, 2012. Graphs, structures, and algorithms.
-  Jaroslav Nešetřil, Patrice Ossona de Mendez, and David R. Wood. Characterisations and examples of graph classes with bounded expansion. European J. Combin., 33(3):350–373, 2012. arXiv:0902.3265.
-  János Pach and Géza Tóth. Graphs drawn with few crossings per edge. Combinatorica, 17(3):427–439, 1997.
-  Felix Reidl, Fernando Sanchez Villaamil, and Konstantinos Stavropoulos. Characterising bounded expansion by neighbourhood complexity. arXiv:1603.09532.
-  Noah Streib and William T. Trotter. Dimension and height for posets with planar cover graphs. European J. Combin., 35:474–489, 2014.
-  William T. Trotter. Combinatorics and partially ordered sets. Johns Hopkins Series in the Mathematical Sciences. Johns Hopkins University Press, Baltimore, MD, 1992. Dimension theory.
-  William T. Trotter. Partially ordered sets. In Handbook of combinatorics, Vol. 1, 2, pages 433–480. Elsevier Sci. B. V., Amsterdam, 1995.
-  William T. Trotter, Jr. and John I. Moore, Jr. The dimension of planar posets. J. Combinatorial Theory Ser. B, 22(1):54–67, 1977.
-  Bartosz Walczak. Minors and dimension. J. Combin. Theory Ser. B, 122:668–689, 2017. arXiv:1407.4066.
-  Mihalis Yannakakis. The complexity of the partial order dimension problem. SIAM J. Algebraic Discrete Methods, 3(3):351–358, 1982.