The crossing number of a graph666We consider graphs that are undirected, simple, and finite. Let and respectively be the vertex and edge sets of . Let and . For each vertex of , let be the neighbourhood of in . The degree of , denoted by , is . When the graph is clear from the context, we write . Let be the maximum degree of . , denoted by , is the minimum number of crossings in a drawing777A drawing of a graph represents each vertex by a distinct point in the plane, and represents each edge by a simple closed curve between its endpoints, such that the only vertices an edge intersects are its own endpoints, and no three edges intersect at a common point (except at a common endpoint). A drawing is rectilinear if each edge is a line-segment, and is convex if, in addition, the vertices are in convex position. A crossing is a point of intersection between two edges (other than a common endpoint). A drawing with no crossings is crossing-free. A graph is planar if it has a crossing-free drawing. of in the plane; see [45, 58] for surveys. The crossing number is an important measure of non-planarity of a graph , with applications in discrete and computational geometry [43, 63], VLSI circuit design [4, 33, 34], and in several other areas of mathematics and theoretical computer science; see  for details. In information visualisation, one of the most important measures of the quality of a graph drawing is the number of crossings [48, 49].
Computing the crossing number is -hard , and remains so for simple cubic graphs [29, 47]. Moreover, the exact or even asymptotic crossing number is not known for specific graph families, such as complete graphs , complete bipartite graphs [38, 51, 53], and cartesian products [1, 6, 25, 52]. On the other hand, Grohe  developed a quadratic-time algorithm that decides whether a given graph has crossing number at most some fixed number , and if this is the case, produces a drawing of the graph with at most crossings. Kawarabayashi and Reed  improved the time complexity to linear.
Given that the crossing number seems so difficult, it is natural to focus on asymptotic bounds rather than exact values. The ‘crossing lemma’, conjectured by Erdős and Guy  and first proved888A remarkably simple probabilistic proof of the crossing lemma with was found by Chazelle, Sharir and Welzl (see ). See [37, 41] for subsequent improvements. by Leighton  and Ajtai et al. , gives such a lower bound. It states that for every graph with average degree greater than has
Upper bounds on the crossing number of general families of graphs have been less studied, and are the focus of this paper. Obviously for every graph . A family of graphs has linear999If the crossing number of a graph is linear in the number of edges then it is also linear in the number of vertices. To see this, let be a graph with vertices and edges. Suppose that . If then and we are done. Otherwise by the crossing lemma. Thus and . crossing number if for some constant and for every graph in the family. The following theorem of Pach and Tóth  and Böröczky et al.  shows that graphs of bounded Euler genus101010Let be a surface. An embedding of a graph in is a crossing-free drawing of in . The Euler genus of equals if is the sphere with handles, and equals if is the sphere with cross-caps. The Euler genus of a graph is the minimum Euler genus of a surface in which there is an embedding of . In what follows, by a face of an embedded graph we mean the set of vertices on the boundary of the face. See  for a thorough treatment of graph embeddings. and bounded degree have linear crossing number.
For every integer , there are constants and , such that every graph with Euler genus has crossing number
In the case of orientable surfaces, Djidjev and Vrťo  greatly improved the dependence on in Theorem 1, by proving that . Wood and Telle  proved that bounded-degree graphs that exclude a fixed graph as a minor111111Let be an edge of a graph . Let be the graph obtained by identifying the vertices and , deleting loops, and replacing parallel edges by a single edge. Then is obtained from by contracting . A graph is a minor of a graph if can be obtained from a subgraph of by contracting edges. A family of graphs is minor-closed if implies that every minor of is in . is proper if it is not the family of all graphs. A deep theorem of Robertson and Seymour  states that every proper minor-closed family can be characterised by a finite family of excluded minors. Every proper minor-closed family is a subset of the -minor-free graphs for some graph . We thus focus on minor-closed families with one excluded minor. have linear crossing number.
Theorem 2 ().
For every graph , there is a constant , such that every -minor-free graph has crossing number
Theorem 2 is stronger than Theorem 1 in the sense that graphs of bounded genus exclude a fixed graph as a minor, but there are graphs with a fixed excluded minor and arbitrarily large genus121212Since the genus of a graph is the sum of the genera of its biconnected components, it is easy to construct graphs with unbounded genus and no -minor. (Take copies of for example.) A more highly connected example is the complete bipartite graph , which has no -minor and has unbounded genus . Seese and Wessel  constructed a family of graphs, each with no -minor and maximum degree , and with unbounded genus. On the other hand, Theorem 1 has better dependence on than Theorem 2. For other work on minors and crossing number see [7, 8, 9, 22, 24, 28, 29, 39, 47].
Note that to obtain a linear upper bound on the crossing number, it is necessary to assume both bounded degree and some structural assumption such as an excluded minor (as in Theorem 2). For example, has no -minor, yet its crossing number is [51, 38]. On the other hand, bounded degree does not by itself guarantee linear crossing number. For example, a random cubic graph on vertices has bisection width [11, 13], which implies that its crossing number is [18, 33].
Pach and Tóth  proved that the upper bound in Theorem 1 is best possible, in the sense that for all and , there is a toroidal graph with vertices and maximum degree whose crossing number is . In Section 2 we extend this lower bound to graphs with no -minor, no -minor, and more generally, with no -minor. Our main result is to prove a matching upper bound for all graphs excluding a fixed minor. That is, we improve the quadratic dependence on in Theorem 2 to linear.
For every graph there is a constant , such that every -minor-free graph has crossing number
While our upper bound in Theorem 3 is optimal in terms of and , it remains open whether every graph excluding a fixed minor has crossing number , as is the case for graphs of bounded Euler genus. Note that a upper bound is stronger than a upper bound; see Section 3. Wood and Telle  conjectured that every graph excluding a fixed minor has crossing number . In Section 4, we establish this conjecture for -minor-free graphs, and prove the same bound on the rectilinear crossing number131313The rectilinear crossing number of a graph , denoted by , is the minimum number of crossings in a rectilinear drawing of . The convex crossing number, denoted by , is the minimum number of crossings in a convex drawing of . of -minor-free graphs. In addition to these results, we establish in Section 5 optimal bounds on the convex crossing number of interval graphs, chordal graphs, and bounded pathwidth graphs.
It is worth noting that our proof is constructive, assuming a structural decomposition (Theorem 24) by Robertson and Seymour  is given. Demaine et al.  gave a polynomial-time algorithm to compute this decomposition. Consequently, our proof can be converted into a polynomial-time algorithm that, given a graph excluding a fixed minor, finds a drawing of with the claimed number of crossings.
2 Lower Bounds
In this section we describe graphs that provide lower bounds on the crossing number. The constructions are variations on those by Pach and Tóth . We include them here to motivate our interest in matching upper bounds in later sections.
For all positive integers and , such that and , there is a (chordal) -minor-free graph with , , and
Start with as the base graph. For each edge of , add new vertices, each adjacent to and . The resulting graph is chordal and -minor-free, , and . Take disjoint copies of to obtain a -minor-free graph on vertices and maximum degree . Thus . To complete the proof, we use a standard technique to prove that . Consider a drawing of with crossings. For each vertex of degree in , contract one of its incident edges to obtain a drawing of a multigraph . This operation does not affect the number of crossings. Thus . Consider an edge between and in that is crossed by the least number of edges in . Redraw each remaining edge between and so close to that the parallel edges between and appear as one edge in the drawing. Again, this does not increase the number of crossings. Repeat this step for each pair of vertices of . The resulting drawing of is equivalent to an optimal drawing of where each pair of crossing edges is replaced by pairs of crossing edges in the drawing of . Since , . ∎
A similar technique gives the following lemma.
For every set of positive integers such that for , there are infinitely many (chordal) -minor-free graphs such that the degree set of is and
For each , let . By Lemma 4, there is a (chordal) -minor-free graph with five vertices of degree and vertices of degree , such that
Every graph created by taking one or more disjoint copies of each of is -minor-free with degree set , and . ∎
The above results generalise to -minor-free graphs, for .
For every integer and every such that for and for , there exists infinitely many -minor-free graphs with and
for some absolute constant . Moreover, is chordal for .
For , use as the starting graph. For , use . The remaining arguments follow the proof of Lemma 4 and use the fact that and . ∎
3 Linear Bounding Functions
The upper bounds on the crossing number that are proved in this paper are linear (in the number of vertices) for graphs with bounded degree, but can be quadratic or more for graphs of unbounded degree. A number of functions satisfy these properties. The smallest such function that we consider is:
Most of the graphs that we consider have a linear number of edges. Thus it is worthwhile to note that the dependence on the maximum degree in (1) is at most linear. In particular,
The next best function is:
For every graph with minimum degree (ignoring isolated vertices),
Since is at least adjacent to ,
Let be a class of graphs closed under taking subdivisions. Suppose that
for every graph . Then
for every graph .
Let . Let be the graph obtained from by subdividing every edge once. By assumption, and
The result follows since . ∎
We can also conclude a bound from for sparse graphs.
Let be a graph such that every subgraph of on vertices has at most edges. Then
For integers , let
Let . Thus
The next best function is:
For every graph ,
Let be an edge of . Thus , implying . Hence
Again note that
4 Drawings Based on Planar Decompositions
Let and be graphs, such that each vertex of is a set of vertices of (called a bag). For each vertex of , let be the subgraph of induced by the bags that contain . Then is a decomposition of if:
is connected and nonempty for each vertex of , and
and touch141414Let and be subgraphs of a graph . Then and intersect if , and and touch if they intersect or and for some edge of . for each edge of .
Let be a decomposition of a graph . The width of is the maximum cardinality of a bag. Let be a vertex of . The number of bags in that contain is the spread of in . The spread of is the maximum spread of a vertex of . A decomposition of is a partition if every vertex of has spread . The order of is the number of bags. has linear order if for some constant . If the graph is a tree, then the decomposition is a tree decomposition. If the graph is a path, then the decomposition is a path decomposition. The decomposition is planar if the graph is planar.
A decomposition of a graph is strong if and intersect for each edge of . The treewidth (pathwidth) of , is less than the minimum width of a strong tree (path) decomposition of . For each constant , the graphs of treewidth at most form a proper minor-closed class. Treewidth is particularly important in structural and algorithmic graph theory; see the surveys [5, 50].
Every graph with a planar partition of width has a rectilinear drawing in which each edge crosses at most other edges. The total number of crossings,
The following drawing algorithm is in . By the Fáry-Wagner Theorem, has a rectilinear drawing with no crossings. Let . Let be the disc of radius centred at each bag of . For each edge of , let be the union of all line-segments with one endpoint in and one endpoint in . For some , we have for all distinct bags and of , and for all edges and of that have no endpoint in common. For each vertex of in bag of , position inside so that is in general position (no three collinear). Draw every edge of straight. Thus no edge passes through a vertex.Suppose that two edges and cross. Then and have distinct endpoints in a common bag, as otherwise two edges in would cross. (The analysis that follows is new.) Say is an endpoint of and is an endpoint of , where is some bag with . Without loss of generality, . Charge the crossing to . The number of crossings charged to is at most
So the total number of crossings is as claimed. ∎
Every graph with no -minor has rectilinear crossing number
We now extend Lemma 11 from planar partitions to planar decompositions.
Suppose that is a planar decomposition of a graph with width , in which each vertex of has spread at most . Then has crossing number
Moreover, has a drawing with the claimed number of crossings, in which each edge is represented by a polyline with at most bends.
For each vertex of , let be a bag of that contains . Think of as the ‘home’ bag of . For each edge of , let be a minimum length path in between and , such that or is in every bag in . Let be the subdivision of obtained by subdividing each edge of once for each internal bag in . Then defines a planar partition of , where each original vertex is in , and each division vertex is in the corresponding bag. We say a division vertex of belongs to and owns , if corresponds to a bag in that contains . If corresponds to a bag that contains both and , then arbitrarily choose or to be the owner of .
Apply the drawing algorithm in Lemma 11 to the planar partition of . We obtain a rectilinear drawing of , which defines a drawing of since is a subdivision of . Each edge of is represented by a polyline with bends, which is at most . We now bound the number of crossings in the drawing of , which in turn bounds the number of crossings in the drawing of .
Let be a total order on such that if then for all .
Say edges and of cross. As proved in Lemma 11, and have distinct endpoints in a common bag . Let and be these endpoints of and respectively. Let and be the vertices of that own and respectively. Without loss of generality, . Charge the crossing to the pair , where is the bag in corresponding to .
Consider a bag in , where . Thus . Consider a vertex . If then edges of are incident to , which is the only vertex in that belongs to . If then there are at most division vertices in that belong to , and there are at most edges of incident to a division vertex in that belongs to (since each division vertex has degree in ). Thus the number of crossings charged to is at most
For each vertex of , since is in at most bags of , the number of crossings charged to some pair is at most . Hence the total number of crossings is at most
Our first application of Lemma 13 concerns graphs of bounded treewidth. Ding and Oporowski  proved that every graph with bounded treewidth has a tree decomposition of bounded width, in which the spread of each vertex is bounded by its degree. In particular:
Theorem 14 (Ding and Oporowski ).
Every graph with treewidth has a tree decomposition of width and order , in which the spread of each vertex is at most .
Every graph with treewidth has crossing number
Lemma 13 also leads to the following upper bound on the crossing number.
Let be a planar decomposition of a graph , such that every bag in is a clique in , and every pair of adjacent vertices in are in at most common bags in . Then
Draw as in the proof of Lemma 13. We now count the crossings in between edges and that have no common endpoint. Each crossing between and can be charged to a bag that contains distinct vertices and , where and . Since is a clique, is an edge of . Charge the crossing to the pair . At most one crossing between and is charged to . Thus at most crossings are charged to . Since and are in at most common bags, the number of crossings charged to is at most . Thus the total number of crossings between edges with no common endpoint is at most . The result follows since it is folklore that equals the minimum, taken over all drawings of , of the number of crossings between pairs of edges of with no endpoint in common; see  for example. ∎
5 Interval Graphs and Chordal Graphs
A graph is chordal if every induced cycle is a triangle. An interval graph is the intersection graph of a set of intervals in . Every interval graph is chordal.
Every interval graph has convex crossing number
Jamison and Laskar  proved that is an interval graph if and only if there is a linear order of such that if and then . Orient the edges of left to right in . Position on a circle in the order of , with the edges drawn straight. Say edges and cross. Without loss of generality, . Thus . Charge the crossing to . Say the out-neighbours of are . The in-neighbourhood of each is a clique including . Hence each has at most in-neighbours to the left of . Now has neighbours to the right of . Thus the number of crossings charged to is at most . Hence the number of crossings charged to outgoing edges at is at most . Therefore the total number of crossings is at most , where is the out-degree of . The other claims follow since . ∎
It is well known that the pathwidth of a graph equals the minimum such that is a spanning subgraph of an interval graph with .
Every graph with pathwidth has convex crossing number
is a spanning subgraph of an interval graph with . Apply the drawing algorithm in the proof of Theorem 17 to . Say edges and of cross. Without loss of generality, . Thus . Charge the crossing to . Now has at most neighbours in to the right of . The in-neighbourhood of is a clique in including . Hence has at most neighbours to the left of . Thus the number of crossings charged to is at most . Since has less than edges, the total number of crossings is at most . ∎
Let be an outerplanar decomposition of a graph . Then has a convex drawing such that if two edges and cross, then some bag of contains both an endpoint of and an endpoint of .
Assign each vertex of to a bag that contains . Fix a crossing-free convex drawing of . Replace each bag of by the set of vertices of assigned to . Draw the edges of straight. Consider two edges and of . Thus there is a path in between and and every bag in contains or . Similarly, there is a path in between and and every bag in contains or . Now suppose that and cross. Without loss of generality, the endpoints are in the cyclic order . Thus in the crossing-free convex drawing of , the vertices appear in this cyclic order. Since is crossing-free, and have a bag of in common. Thus contains or , and or . ∎
Every chordal graph has convex crossing number
It is well known that every chordal graph has a strong tree decomposition in which each bag is a clique . Since trees are outerplanar, by Lemma 19, has a convex drawing such that if two edges and of cross, then some bag of contains or , and or . Say contains and . Since is a clique, is an edge. Charge the crossing to . In every crossing charged to , one edge is incident to and the other edge is incident to . Since edges are drawn straight, no two edges cross twice. Thus the number of crossings charged to is at most . Hence the total number of crossings is as claimed. ∎
Note that the upper bound in Theorem 20 is within a constant factor of being tight for sufficiently dense chordal graphs for which all the vertex degrees are within a constant factor of each other. In particular, consider a chordal graph in which every vertex degree is within a constant factor of . Then the upper bound in Theorem 20 is , and the crossing lemma gives a lower bound of (assuming ).
Also note that there are chordal graphs for which the lower bound is far from the crossing number. Let be the split graph obtained from a triangle by adding a set of independent vertices, each adjacent to every vertex in . Then but since contains as a subgraph.
Every chordal graph with no -clique has convex crossing number
6 Excluding a Fixed Minor
In this section we prove our main result (Theorem 3): for every graph there is a constant , such that every -minor-free graph has a crossing number at most . The proof is based on Robertson and Seymour’s rough characterisation of -minor-free graphs, which we now introduce. For an integer and a surface , Robertson and Seymour  defined a graph to be -almost embeddable in if has a set of at most vertices (called apices) such that can be written as such that:
has an embedding in .
The graphs (called vortices) are pairwise disjoint.
There are faces151515Recall that we identify a face with the set of vertices on its boundary. of the embedding of in , such that for .
For , if in clockwise order about the face, then has a strong -path decomposition of width at most , such that each vertex is in the -th bag of .
For all integers and , there is a constant , such that every graph that is -almost embeddable in some surface with Euler genus at most , has crossing number at most .
Let and be the parts of as specified in the definition of -almost embeddable graph. Let and . Start with an embedding of in . For each , draw vortex inside of the face on , as prescribed in Theorem 18. Then the resulting drawing of in has at most crossings. Replace each crossing by a dummy degree- vertex. The resulting graph is embedded in By Theorem 1, for some constant ,
Since , we conclude that
By Lemma 23 below,
In the -minor-free graph , the number of edges is at most , for some constant (see ). Thus . ∎
For every vertex of a graph ,
Pach and Tóth  proved that for every edge of a graph ,
The claim follows by applying this result to each edge incident to . ∎
Let and be disjoint graphs. Suppose that and are cliques of and respectively, each of size , for some integer . Let and