Minimizing the number of edge-crossings in a graph drawing in the plane (the crossing number of the graph, cf. Definition 2) is considered one of the most important attributes of a “nice drawing” of a graph, and this question has found numerous other applications (for example, in VLSI design  and in discrete geometry ). Consequently, a great deal of research work has been invested into understanding what forces the graph crossing number to be high. There exist strong quantitative lower bounds, such as the famous Crossing Lemma [1, 12]. However, the quantitative bounds show their strength typically in dense graphs, and hence they do not shed much light on the structural properties of graphs of high crossing number.
The reasons for sparse graphs to have many crossings in any drawing are structural – there is a lot of “nonplanarity” in them. These reasons can be understood via corresponding minimal obstructions, the so called -crossing-critical graphs (cf. Section 2 and Definition 2), which are the subgraph-minimal graphs that require at least crossings. There are only two -crossing-critical graphs without degree- vertices, the Kuratowski graphs and , but it has been known already since Širáň’s  and Kochol’s  constructions that the structure of -crossing-critical graphs is quite rich and non-trivial for any . Already the first nontrivial case of shows a dramatic increase in complexity of the problem. Yet, Bokal, Oporowski, Richter, and Salazar recently succeeded in obtaining a full description  of all the -crossing-critical graphs up to finitely many “small” exceptions.
To our current knowledge, there is no hope of extending the explicit description from  to any value . We, instead, give for any fixed positive integer an asymptotic structural description of all sufficiently large -crossing-critical graphs.
We refer to subsequent sections for the necessary formal concepts. On a high level of abstraction, our contribution can be summarized as follows:
There exist three kinds of local arrangements—a crossed band of uniform width, a twisted band, or a twisted fan—such that any optimal drawing of a sufficiently large -crossing-critical graph contains at least one of them.
There are well-defined local operations (replacements) performed on such bands or fans that can reduce any sufficiently large -crossing-critical graph to one of finitely many base -crossing-critical graphs.
A converse—a well-defined bounded-size expansion operation—can be used to iteratively construct each -crossing-critical graph from a -crossing-critical graph of bounded size. This yields a way to enumerate all the -crossing-critical graphs of at most given order in polynomial time per each generated graph. More precisely, the total runtime is times the output size.
To give a closer (but still informal) explanation of these points, we should review some of the key prior results. First, the infinite -crossing-critical family of Kochol  explicitly showed one basic method of constructing crossing-critical graphs—take a sequence of suitable small planar graphs (called tiles, cf. Section 3), concatenate them naturally into a plane strip and join the ends of this strip with the Möbius twist. See Figure 1. Further constructions of this kind can be found, e.g., in [2, 14, 16]. In fact,  essentially claims that such a Möbius twist construction is the only possibility for ; there, the authors give an explicit list of tiles which build in this way all the -crossing-critical graphs up to finitely many exceptions.
The second basic method of building crossing-critical graphs was invented later by Hliněný ; it can be roughly described as constructing a suitable planar strip whose ends are now joined without a twist (i.e., making a cylinder), and adding to it a few edges which then have to cross the strip. See again Figure 1 for an illustration. Furthermore, diverse crossing-critical constructions can easily be combined together using so called zip product operation of Bokal  which preserves criticality. To complete the whole picture, there exists a third, somehow mysterious method of building -crossing-critical graphs (for sufficiently high values of ), discovered by Dvořák and Mohar in . The latter can be seen as a degenerate case of the Möbius twist construction, such that the whole strip shares a central high-degree vertex, and we skip more details till the technical parts of this paper.
As we will see, the three above sketched construction methods roughly represent the three kinds of local arrangements mentioned in point (1). In a sense, we can thus claim that no other method (than the previous three) of constructing infinite families of -crossing-critical graphs is possible, for any fixed . Moving on to point (2), we note that all three mentioned construction methods involve long (and also “thin”) planar strips, or bands as subgraphs (which degenerate into fans in the third kind of local arrangements; cf. Definition 2). We will prove, see Corollary 3.2, that such a long and “thin” planar band or fan must exist in any sufficiently large -crossing-critical graph, and we analyse its structure to identify elementary connected tiles of bounded size forming the band. We then argue that we can reduce repeated sections of the band while preserving -crossing-criticality. Regarding point (3), the converse procedure giving a generic bounded-size expansion operation on -crossing-critical graphs is described in Theorem 4.3 (for a quick illustration, the easiest case of such an expansion operation is edge subdivision, that is replacing an edge with a path, which clearly preserves -crossing-criticality).
After giving the definitions and preliminary results about crossing-critical graphs in Section 2, we show a new structural characterisation of plane graphs of bounded path-width which forms the cornerstone of our paper in Section 3. Then, in Section 4, we deal with the structure and reductions / expansions of crossing-critical graphs, presenting our main results. Some final remarks are presented in Section 5.
2 Graph drawing and the crossing number
In this paper, we consider multigraphs by default, even though we could always subdivide parallel edges (with a slight adjustment of definitions) in order to make our graphs simple. We follow basic terminology of topological graph theory, see e.g. .
A drawing of a graph in the plane is such that the vertices of are distinct points and the edges are simple curves joining their end vertices. It is required that no edge passes through a vertex, and no three edges cross in a common point. A crossing is then an intersection point of two edges other than their common end. A drawing without crossings in the plane is called a plane drawing of a graph, or shortly a plane graph. A graph having a plane drawing is planar.
The following are the core definitions of our research. [crossing number] The crossing number of a graph is the minimum number of crossings of edges in a drawing of in the plane.
[crossing-critical] Let be a positive integer. A graph is -crossing-critical if , but every proper subgraph of has .
Furthermore, suppose is a graph drawn in the plane with crossings. Let be the plane graph obtained from this drawing by replacing the crossings with new vertices of degree . We say that is the plane graph associated with the drawing, shortly the planarization of , and the new vertices are the crossing vertices of .
Structural properties of crossing-critical graphs have been studied for more than two decades, and we now briefly review some of the previous important results which we shall use. First, we remark that a -crossing-critical graph may have no drawing with only crossings (examples exist already for ). Richter and Thomassen  proved the following upper bound: [] Every -crossing-critical graph has a drawing with at most crossings.
Interestingly, although the bound of Theorem 2 sounds rather weak and we do not know any concrete examples requiring more than crossings, the upper bound has not been improved for more than two decades. We not only use this important upper bound, but also hope to be able to improve it in the future using our results.
Our approach to dealing with “long and thin” subgraphs in crossing-critical graphs relies on the folklore structural notion of path-width of a graph, which we recall in Definition 3.1. Hliněný  proved that -crossing-critical graphs have path-width bounded in terms of , and he and Salazar  showed that -crossing-critical graphs can contain only a bounded number of internally disjoint paths between any two vertices. [] Every -crossing-critical graph has path-width (cf. Definition 3.1) at most .
Another useful concept for this work is that of nests in a drawing of a graph (cf. Definition 2), implicitly considered already in previous works [7, 8], and explicitly defined by Hernandez-Velez et al.  who concluded that no optimal drawing of a -crossing-critical graph can contain a -, -, or -nest of large depth compared to .
Lastly, we remark that by trivial additivity of the crossing number over blocks, we may (and will) restrict our attention only to -connected crossing-critical graphs. We formally argue as follows. For , let us say a graph is -crossing-critical if it has crossing number exactly and all proper subgraphs have crossing number at most .
A graph is -crossing-critical if and only if there exist positive integers and such that , has exactly -connected blocks , …, , and the block is -crossing-critical for .
Hence, strictly respecting Proposition 2, we should actually study -connected -crossing-critical graphs. To keep the presentation simpler, we stick with -crossing-critical graphs, but we remark that our results also hold in the more refined setting.
3 Structure of plane tiles
The proof of our structural characterisation of crossing-critical graphs can be roughly divided into two main parts. The first one, presented in this section, establishes the existence of specific plane bands (resp. fans) and their tiles in crossing-critical graphs. The second part will then, in Section 4, closely analyse these bands and tiles. Unlike a more traditional “bottom-up” approach to tiles in crossing number research (e.g., ), we define tiles and deal with them “top-down”, i.e., describing first plane bands or fans and then identifying tiles as their small elementary parts. Our key results are summarized below in Theorem 3.1 and Corollary 3.2.
[band and fan] Let be a -connected plane graph. Let and be distinct faces of and let , and be some of the vertices incident with and , respectively, listed in the cyclic order along the faces. If , …, are pairwise vertex-disjoint paths in such that joins with , for , then we say that forms an -band of length . Note that may consist of only one vertex .
Let and be as above. If is a vertex of and , …, are paths in such that joins with , for , and the paths are pairwise vertex-disjoint except for their common end , then we say that forms an -fan of length . The -fan is proper if is not incident with .
[tiles and support] Let be either an -band or an -fan of length . For , let be an arc between and drawn inside , and let be an arc drawn between and in in the case of the band; are null when we are considering a fan. Furthermore, choose the arcs to be internally disjoint. Let be the closed curve consisting of , , , and . Let be the connected part of the plane minus that contains none of the paths () in its interior.
The subgraphs of drawn in the closures of , …, are called tiles of the band or fan (and the tile of includes by this definition). The union of these tiles is the support of the band or fan. The union of the arcs is the -span of the band or fan, and in the case of a band, the union of the arcs is the -span of the band.
[nests] Let be a -connected plane graph. For an integer , a -nest in of depth is a sequence of pairwise edge-disjoint cycles such that for some set of vertices and for every , the cycle is drawn in the closed disk bounded by and .
Let be a face of and let , , …, be some of the vertices incident with listed in the cyclic order along the face. Let , …, be pairwise vertex-disjoint paths in such that joins with , for . Then, we say that forms an -nest of depth . Similarly, let , , …, , be some of the vertices incident with , let , …, be paths in such that joins with , for , and the paths intersect only in . Then, we say that form a degenerate -nest of depth . See Figure 3. Note that degenerate -nests are the same as non-proper -fans.
3.1 Plane graphs of bounded path-width
Our cornerstone claim, interesting on its own, is a structure theorem for plane graphs of bounded path-width. Before stating it, we recall the definition of path-width. [path decomposition] A path decomposition of a graph is a pair , where is a path and is a function that assigns subsets of , called bags, to nodes of such that
for each edge , there exists such that , and
for every , the set induces a non-empty connected subpath of .
The width of the decomposition is the maximum of over all vertices of , and the path-width of is the minimum width over all path decompositions of .
Let denote the first node and the last node of in a path decomposition . For , let be the node of preceding , and let . For , let be the node of following , and let . The path decomposition is proper if for all distinct . The interior width of the decomposition is the maximum over over all nodes of distinct from and . The path decomposition is -linked if for all and contains vertex-disjoint paths from to . The order of the decomposition is .
Let , , and be non-negative integers, and be an arbitrary non-decreasing function. There exist integers and such that the following holds. Let be a -connected plane graph and let be a set of at most vertices of of degree at most . If has path-width at most and , then one of the following holds:
contains a -nest, a -nest, a -nest, an -nest, or a degenerate -nest for some face of , of depth , and with all its cycles or paths disjoint from , or
for some , contains an -band or a proper -fan (where and are distinct faces and is a vertex) of length at least and with support disjoint from , such that each of its tiles has size at most .
We pay close attention to explaining Theorem 3.1, because of its great importance in this paper. Comparing it to Definition 3.1, one may think that there is not much difference—the bags of a path decomposition of of width at most might perhaps play the role of tiles of the band or fan in the second conclusion. Unfortunately, this simple idea is quite far from the truth. The subgraphs induced by the bags may not be “drawn locally”, that is, its edges may be geometrically far apart in the plane graph . As an example, consider the width path decomposition of a cycle where one of the vertices of the cycle appears in all the bags.
The main message of Theorem 3.1 thus is that in a plane graph of bounded path-width we can find a long band which is “drawn locally” and decomposes into well-defined small and connected tiles (cf. Definition 2). Otherwise, such a graph must contain some kind of a deep nest or fan. However, as we will see in Corollary 3.2, the latter structures are impossible in the planarizations of optimal drawings of crossing-critical graphs.
The proof of Theorem 3.1 requires some preparatory work, and it uses tools of structural graph theory and of semigroup theory in algebra, which we present now.
Let and be non-negative integers, and let be an arbitrary non-decreasing function. There exist integers and such that the following holds. If a graph has a proper path decomposition of interior width at most , adhesion at most , and order at least , then for some and , also has a -linked proper path decomposition of interior width at most and order at least .
Let be a proper path decomposition of of interior width at most and adhesion at most . We prove the claim by induction on . If , then is -linked, and thus the claim holds with and . Hence, assume that . Let and be the integers from the statement of the lemma for and the interior width bounded by . Let and .
We say that a node of distinct from its endpoints is unbroken if and contains pairwise vertex-disjoint paths from to , and broken otherwise. By Menger’s theorem, if is broken, then there exist sets such that , , , , and there is no edge from to in . If contains a subpath of consecutive unbroken nodes, then the restriction of to is an -linked proper path decomposition of interior width at most and order at least . Otherwise, one of each consecutive nodes of is broken, and thus has a proper path decomposition of interior width at most , adhesion at most , and order at least . Hence, the claim follows by the induction hypothesis. ∎
A crucial technical step in the proof of Theorem 3.1 is to analyse a topologigal structure of the bags of a path decomposition of a plane graph , and to find many consecutive subpaths of on which the decomposition repeats the same “topological behavior”. For this we are going to model the bags of the decomposition as letters of a string over a suitable finite semigroup (these letters present an abstraction of the bags), and to apply the following algebraic tool, Lemma 3.1.
Applying Simon’s factorisation forest.
Let be a rooted ordered tree (i.e., the order of children of each vertex is fixed). Let be a function that to each leaf of assigns a string of length , such that for each non-leaf vertex of , is the concatenation of the strings assigned by to the children of in order. We say that yields the string assigned to the root of by . If the letters of the string are elements of a semigroup , then for each , let denote the product of the letters of in . Recall that an element of is idempotent if . A tree is an -factorization tree if for every vertex of with more than two children, there exists an idempotent element such that for each child of (and hence also ). Simon  showed existence of bounded-depth -factorization trees for every string; the improved bound in the following lemma was proved by Colcombet :
[] For every finite semigroup and each string of elements of , there exists an -factorization tree of depth at most yielding this string.
We will combine Lemma 3.1 with the following observation. Let be an arbitrary non-decreasing function and let be a positive integer. There exist integers and such that if is a rooted tree of depth at most with at least leaves, then for some , there exists a vertex of that has at least children, and the subtree of rooted at each child of has at most leaves.
We prove the claim by induction on . For , it suffices to set and . Suppose that and the claim holds for , with and playing the role of and . Let and . If the subtree rooted at some child of the root has at least leaves, then the claim follows by the induction hypothesis applied to this subtree. Otherwise, the root has at least children, and the subtree rooted in each of them has at most leaves. Hence, we can let be the root and . ∎
Combining Lemmas 3.1 and 3.1, we obtain the following. Let be a non-negative integer and let be an arbitrary non-decreasing function. There exist integers and such that if is a semigroup of order at most and is a string of elements of of length at least , then is a concatenation of strings , , …, , for some integer , such that
for some with , the strings , …, have length at most and
the product of elements of in each of the strings , …, is the same idempotent element of .
We further need to formally define what we mean by a “topological behavior” of bags and subpaths of a path decomposition of our . This will be achieved by the following term of a -type.
In this context we consider multigraphs (i.e., with parallel edges and loops allowed – each loop contributes to degree of the incident vertex, and not necessarily connected) with some of its vertices labelled by distinct unique labels. A plane multigraph is irreducible if has no faces of size or , and every unlabelled vertex of degree at most is an isolated vertex incident with one loop (this loop, hence, cannot bound a -face).
Two plane multigraphs and with some of the vertices labelled are homeomorphic if there exists a homeomorphism of the plane mapping onto so that for each vertex , the vertex is labelled iff is, and then and have the same label. For with some of its vertices labelled using the labels from a finite set , the -type of is the set of all non-homeomorphic irreducible plane multigraphs labelled from and with at most unlabelled vertices, and whose subdivisions are homeomorphic to subgraphs of . Note that for every finite set of labels and every integer , there exist only finitely many irreducible non-homeomorphic plane multigraphs that are labelled with labels from and have at most unlabelled vertices.
The definition of a -type is going to be applied to graphs induced by subpaths of a path decomposition above (Lemma 3.1).
Let be a plane graph and let be its -linked path decomposition. Let and be the endpoints of . Fix pairwise vertex-disjoint paths , …, between and . Consider a subpath of , and let be the subgraph of induced by . If and are the (left and right) endpoints of , we define and . Let us label the vertices of using (some of) the labels as follows: For , let and be the vertices in which intersects and , respectively. If , we give the label and the label . Otherwise, we give the label . For an integer , the -type of is the -type of with this labelling. If contains just one node , then we speak of the -type of .
The -types of subpaths of a linked path decomposition naturally form a semigroup with concatenation of the subpaths, as detailed next.
Let be the set of -types of subpaths of , together with a special element . Let be defined as follows. If for , there exist paths such that the first node of immediately follows the last node of in , the -type of is , and the -type of is , then is defined as the -type of the path obtained from and by adding the edge of joining them. For any other , we define .
is a semigroup.
As associativity is obvious, it suffices to observe that if and are any other consecutive subpaths of with -types and , the path obtained by joining with via an edge of also has -type . ∎
Applying Corollary 3.1, we have the following. Let be an arbitrary non-decreasing function and let and be non-negative integers. There exist integers and such that the following holds. Let be a plane graph and let be its -linked path decomposition of order at least . Then can be split by removal of its edges into subpaths , , …, in order for some integer , such that
there exists with such that each of the paths , …, has length at most , and
the -type of each of the paths , …, is the same idempotent element in the semigroup .
Deconstructing plane graphs of bounded path-width.
A path decomposition of is a coarsening of if and can be expressed as a concatenation of paths , …, such that for . For a subpath , the restriction of the decomposition to is the coarsening of such that for all nodes of distinct from its endpoints.
If is a subgraph of a graph , a -bridge of is either an edge of not belonging to and with both ends in , or a connected component of together with all the edges from this component to .
Let and be non-negative integers, and let be an arbitrary non-decreasing function. There exist integers and such that, for any plane graph that has a proper path decomposition of interior width at most and order at least , the following holds. For some and , also has a -linked proper path decomposition of interior width at most and order at least , such that for each node of distinct from its endpoints, the -type of is the same idempotent element of the semigroup .
In other words, we can find a decomposition in which all topological properties of the drawing that hold in one bag repeat in all the bags. So, for example, if for some node , the vertices of are separated in the drawing from vertices of by a cycle contained in the bag of , then this holds in every bag, and we conclude that the drawing contains a large -nest. Other outcomes of Theorem 3.1 naturally correspond to other possible local properties of the drawings of the bags, and so we are ready to finish the main proof now.
Proof of Theorem 3.1.
Let , and let and be the corresponding integers from Theorem 3.1 applied with . Let and .
Since has path-width at most and , has a proper path decomposition of (interior) width at most and order at least . By Theorem 3.1, there exist integers and such that has a -linked proper path decomposition of interior width at most and order at least , such that for each node of distinct from its endpoints, the -type of is the same idempotent element. Note that the chosen labelling used to define the -type determines which vertices belong to ; hence, there exists such that for all nodes of distinct from its endpoints.
Since , there exists a restriction of this path decomposition of order at least , such that if and are the endpoints of , all vertices of belong to . Let and be the neighbors of and in , respectively, and let (so, has ends and ). For any , we have .
Let be a connected component of the graph . This graph is non-null, since the path decomposition is proper. Let be the induced subgraph of consisting of and all vertices of that have a neighbor in . Let be the path decomposition of with for each . Note that with respect to the drawing of inherited from , all the nodes of have the same idempotent -type. By idempotency, has the same 2-type. Let . Since is connected, every vertex in is connected in to . Since the 2-type of is the same as the 2-type of , any two vertices of are connected by a sequence of paths in with internal vertices in , which implies that is connected. Similarly, each vertex of has a neighbor in . Since the decomposition has order at least and for all non-consecutive nodes and of , it follows that all vertices in have degree at least in , and thus they do not belong to . Consequently, for all .
If , then since has at least nodes forming an independent set in , we conclude that contains paths between any two vertices in . The -th path goes through , so they are pairwise disjoint except for their endpoints, and disjoint from . Such paths form a -nest of depth in . Hence, we can assume that . Since is -connected, it follows that and .
Let and be the neighbors of and in , respectively. Consider a node of distinct from , , , and . Note that the subgraph is connected and vertex-disjoint from . Let denote the face of in which is drawn. Similarly, let denote the face of in which is drawn. Suppose that . Then there exists a cycle in the drawing of separating from . Note that separates from and is disjoint from both of these sets. The existence of such a separating cycle is determined by the -type of , and by the idempotency, has the same -type. Consequently, we can actually choose to be disjoint from and (and thus to separate these sets). But then the cycles for intersect at most in a vertex of , and they form a -nest or a -nest of depth at least .
Hence, we can assume that for each . Let . Since the path decomposition is -linked, the path decomposition of is -linked, via some paths , …, . For , let be the subpath of between its vertices in and . Consider the subgraph . We can order the paths so that has faces with contained in the intersection of the boundaries of and for , where . Note that each -bridge of is a subgraph of for some and is drawn inside one of the faces , …, .
Consider any , and let denote the subpath of between and , for . We may assume that . Then, since is connected, it contains a path between and drawn inside the face of the subgraph for each . Since does not contain a cycle separating from , it follows that does not contain any path between and drawn inside the face of the subgraph . Consequently, for every -bridge of drawn in , there exists and such that . It follows that has a face such that for each and , the endpoints of are incident with . Since intersects the rest of only in and , we conclude that has (not necessarily distinct) faces such that for each , the endpoints of are incident with and the endpoints of are incident with .
Let be a path in starting with a vertex of incident with , ending with a vertex of incident with , and otherwise disjoint from the boundaries of and . Let and . Let be a subpath of (which exists since the decomposition has order at least . For , we let .
If , then all the paths , …, contain the vertex of , and their subpaths between and one of the faces or —say —form an -fan or a degenerate -nest. If , then the paths are pairwise vertex-disjoint and form either an -nest (if ) of depth at least , or an -band (if ). The length of the fan or band obtained this way is , and each of its tiles is a subgraph of for some . This implies that the number of vertices in each tile is at most . Furthermore, as we argued before, the tiles are disjoint from . This completes the proof. ∎
3.2 Consequences for crossing-critical graphs
Let be a positive integer, and let be an arbitrary non-decreasing function. There exist integers and such that the following holds. Let be a -connected -crossing-critical graph, and let be the planarization of a drawing of with the smallest number of crossings. Let denote the set of crossing vertices of . If , then contains an -band or a proper -fan for some distinct faces and or a vertex , such that for some , all the tiles of the band or fan have size at most and are disjoint from , and the length of the band or fan is at least .