Graph saturation problems concern the study of edge-extremal -vertex graphs under various restrictions. They originate in the works of Turán  and Erdős, Hajnal, and Moon . For a family of graphs, a graph without loops or parallel edges is called -saturated when no subgraph of belongs to and for every , where , some subgraph of the graph belongs to . Turán  described, for each , the -vertex graphs that are -saturated and have the maximum number of edges—this led to the introduction of the Turán Numbers where the setting moves from graphs to hypergraphs, see, e.g., the surveys [30, 22]. Analogously, Erdős, Hajnal, and Moon  studied the -vertex graphs that are -saturated and have the minimum number of edges. This sparsest saturation view has also received much subsequent study (see, e.g., ), and our work fits into this latter direction but concerns “drawings of (multi-)graphs” (aka topological (multi-)graphs).
There has been increasing interest in saturation problems on drawings of (multi-)graphs in addition to the abstract graphs above. A drawing is a graph together with a cyclic order of edges around each vertex and the sequence of crossings along each edge so that it can be realized in the plane (or on another specified surface). The saturation conditions usually concern the crossings (which can be thought of as avoiding certain topological subgraphs). The majority of work has been on Turán-type results regarding the maximum number of edges which can occur in an -vertex drawing (without loops and homotopic parallel edges) of a particular drawing style, e.g., -vertex planar drawings are well known to have at most edges for any . In the case of planar drawings (i.e., crossing-free in the plane), the sparsest saturation version (as in Erdős, Hajnal, and Moon ) is also equal to the Turán version: Every saturated planar drawing has edges.
However, for drawing styles that allow crossings in a limited way, these two measures become non-trivial to compare and can indeed be quite different. This interesting phenomenon happens for example for -planar drawings where at most crossings on each edge are allowed; and which are the focus of the present paper. The left of Figure 1 depicts a drawing of the -cycle in which each edge is crossed exactly four times and one cannot add a ninth (non-loop) edge to the drawing while maintaining -planarity, i.e., this is a saturated -planar drawing of . On the other hand, note that even the complete graph in fact admits -planar drawings as shown in the middle of Figure 1.
In this sense, we call a drawing that attains the Turán-type maximum number of edges a max-saturated111Sometimes these drawings are called optimal in the literature . drawing, while a sparsest saturated drawing is called min-saturated.
The target of this paper is to determine the number of edges in min-saturated -planar drawings of loopless multigraphs for , i.e., the smallest number of edges among all saturated -planar drawings with vertices. The answer will always be of the form . However, it turns out that the precise value of depends on numerous subtleties of what precisely we allow in the considered -planar drawings. Such subtleties are formalized by drawing styles , each one with its own constant . As we always require -planarity, we omit from the notation .
For example, restricting to connected graphs, we immediately have at least edges on vertices, i.e., . And in fact we also have for any and all as testified by entangled drawings of cycles like in the left of Figure 1. Allowing disconnected graphs but restricting to contiguous drawings, we immediately have since we have minimum degree at least in that case. And again we also have for all as one can find saturated -planar drawings of matchings like in the right of Figure 1. Other subtleties occur when we distinguish whether selfcrossing edges, repeatedly crossing edges, crossing incident edges etc. are allowed or forbidden. We enable a concise investigation of all possible combinations by first deriving lower bounds on for any drawing style that satisfies only some mild assumptions. We can then consider each drawing style and swiftly determine the exact value of , thus determining the smallest number of edges among all -planar drawings of that style on vertices. Our results are summarized in Table 1.
We first discuss related work in Section 1.1. We formally define saturated drawings, crossings restrictions, and drawing styles in Section 1.2. In Section 2 we introduce the notion of filled drawings and develop a general framework of how to determine lower bounds on the minimum number of edges in such filled drawings. In Section 3 we then prove that the considered saturated drawings are filled drawings and use the said framework to prove the exact bounds in our main theorem. Finally, we conclude in Section 4 by discussing the possibilities and limitations of our approach, also mentioning open problems.
1.1 Related Work
For -planar graphs the Turán-type question, the edge count in max-saturated drawings, is well studied. Any -planar simple222A drawing is simple if any two edges share at most one point. In particular there are no parallel edges. drawing on vertices contains at most edges , and better (and tight) bounds are known for small [1, 26, 27]. Specifically -planar drawings contain at most edges which is tight . For , any -planar drawing with the fewest crossings (among all -planar drawings of the abstract graph) is necessarily simple . Therefore the tight bounds for also hold for drawings that are not necessarily simple. However, already for , Schaefer [29, p. 58] has constructed -planar graphs having no -planar simple drawings, and these easily generalize to all . Pach et al.  conjectured that for every there is a max-saturated -planar graph with a simple -planar drawing. For , the max-saturated -planar homotopy-free multigraphs have been characterized .
In the sparsest saturation setting not only min-saturated -planar drawings are of interest but also min-saturated -planar (abstract) graphs: sparse -planar graphs that are no longer -planar after adding any edge. Brandenburg et al.  and independently Eades et al.  constructed saturated -planar -vertex graphs with only edges and saturated -planar drawings with edges. Barát and Tóth  show that any saturated -planar -vertex drawing () has at least edges, but they remark that their bounds seem suboptimal. Auer et al.  construct saturated -planar graphs with roughly edges and saturated -planar drawings with edges.
Recently, the case of saturation problems for simple drawings has come into focus. The Turán-type question is trivial here as all complete graphs have simple drawings. However, knowing when a given simple drawing is saturated turns out to be rather complex as it has recently been shown that it is NP-complete to decide whether a given simple drawing is saturated . In fact, it is even NP-complete to decide whether a single edge can be inserted into a simple drawing . Contrary to the simple drawings of complete graphs, there are constructions of saturated simple drawings (and generalizations thereof) with only edges [19, 24]. The Turán-type question was studied also for simple drawings of multigraphs [21, 28] where the results distinguish between various drawing restrictions.
In the case of -quasiplanar333A drawing is -quasiplanar if every -set of edges contains a pair of edges that do not cross each other. graphs the focus has been on the Turán-type question. It was conjectured that, for every fixed , every -quasiplanar graph has edges. This has been verified for , but the best general upper bound for simple -quasiplanar graphs is for some constant and has been improved slightly for some special cases . For -simple -quasiplanar graphs a general bound is known where is a constant depending on and and is the inverse of the Ackermann function . Seemingly the only min-saturation results involving this crossing restriction concern so-called outer or convex drawings in which all of the vertices occur on the boundary of a single face of the drawing. Here, as in planar drawings, the min-saturated drawings and max-saturated outer -quasiplanar -vertex drawings coincide [12, 14, 25], and have edges .
Also for the concept of gap-planarity , which generalizes the notion of -planarity, the focus so far has been on the Turán-type question.
1.2 Drawings, Crossing Restrictions, and Drawing Types
Throughout we consider topological drawings in the plane, that is, vertices are represented by distinct points in and edges are represented by continuous curves connecting their respective endpoints. We allow parallel edges but forbid loops. As usual, edges do not pass through vertices, any two edges have only finitely many interior points in common, each of which is a proper crossing, and no three edges cross in a common point. An edge may cross itself but it uses any crossing point at most twice. Also, each of these selfcrossings are counted twice when considering the number of times that edge is crossed.
The planarization of a drawing is the planar drawing obtained from by making each crossing into a new vertex, thereby subdividing the edges involved in the crossing. In a drawing, an edge involved in at least one crossing is a crossed edge, while those involved in no crossing are the planar or uncrossed edges. The cells of a drawing are the connected components of the plane after the removal of every vertex and edge in . In other words, the cells of are the faces of its planarization. A vertex is incident to a cell if is contained in the closure of , i.e., one could at least start drawing an uncrossed edge from into cell .
Two distinct parallel edges and in a drawing are called homotopic, if there is a homotopy of the sphere between and , that is, the curves of and can be continuously deformed into each other along the surface of the sphere while all other vertices of are treated as holes.
In what follows, we investigate drawings that satisfy a specific set of restrictions, where we focus on those with frequent appearance in the literature:
-planar: Each edge is crossed at most times.
H homotopy-free: No two distinct parallel edges are homotopic.
M single-crossing: Any pair of edges crosses at most once and any edge crosses itself at most once (edges with common endpoints have at most common points).
I locally starlike444In other papers this is also called star simple or semi simple and may not allow selfcrossing edges.: Incident edges do not cross (while selfcrossing edges are allowed).
S selfcrossing-free: No edge crosses itself.
branching: The drawing is M single-crossing, I locally starlike, S selfcrossing-free, and H homotopy-free.
A drawing style is just a class of drawings, i.e., a predicate whether any given drawing is in or not. A drawing style is monotone if removing any edge or vertex from any drawing results again in a drawing , i.e., is closed under edge/vertex removal.
We consider drawing styles given by all -planar drawings of finite, loopless multigraphs obeying a subset of the restrictions above. Such a drawing style is denoted by . We focus on the restrictions M forbidding multicrossings, S forbidding selfcrossings, I forbidding incident crossings, and H forbidding homotopic edges. Note that the -planar drawing style is monotone, and so is for each . However, the style of all homotopy-free drawings is not monotone, as removing a vertex may render two edges homotopic.
We are interested in -planar drawings in to which no further edge can be added without either violating -planarity or any of the restrictions in , and particularly in how sparse these drawings can be; namely, the sparsest saturated such drawings.
A drawing is -saturated for drawing style if and the addition of any new edge to results in a drawing .
In order to determine the sparsest -planar -saturated drawings for restrictions in , we introduce in Section 2 the concept of filled drawings in general monotone drawing styles and give lower bounds on the number of edges in these. Using the lower bounds for filled drawings and constructing particularly sparse -saturated drawings, we then give in Section 3 the precise answer for all and for the branching style, i.e., , leaving open only a few cases for . Our results are summarized in Table 1 and formalized in Section 3. Finally, in Section 4 we discuss where our approaches fail when but .
|minimum number of|
|restrictions||edges of saturated -planar||tight example555To attain the stated bound via these constructions, insert an isolated vertex in each empty cell.|
|drawings on vertices|
|no restriction||Figure 2|
|I no incident crossings|
|S no selfcrossings||Figure 3|
|S no self- and I no incident crossings|
|M no multicrossings||Figure 4|
|S no self- and M no multicrossings||Figure 5|
|I no incident and M no multicrossings||Figure 6, Figure 7|
|S no self-, M no multi-,||Figure 1 (right), Figure 8|
|and I no incident crossings|
|S no self-, M no multi-, I no incident|
|crossings, and H homotopy-free|
2 Lower Bounds and Filled Drawings
Throughout this section, let be an arbitrary monotone drawing style; not necessarily -planar or defined by any of the restrictions in Section 1.2. Recall that is monotone if it is closed under the removal of vertices and/or edges.
A drawing is filled if any two vertices that are incident to the same cell of are connected by an uncrossed edge that lies completely in the boundary of .
For example, the filled crossing-free homotopy-free drawings are exactly the planar drawings of loopless multigraphs with every face bounded by three edges. Using Euler’s formula, such drawings on vertices have exactly edges. In this section we derive lower bounds on the number of edges in -vertex filled drawings in drawing style .
Another important example of filled drawings are those in which every cell has at most one incident vertex. Note that every cell in a filled drawing has at most three incident vertices. Generally, for a drawing we use the following notation:
For a drawing , let be its graph and be its planarization. A component of is a connected component of . A cut-vertex of is a cut-vertex of that is also a cut-vertex of . And finally, is essentially -connected if it has at least one edge and no cut-vertex. So, a drawing is essentially -connected if and only if one component has at least one edge, all other components are isolated vertices and along the boundary of each cell each vertex appears at most once. This means that for each simple closed curve that intersects in exactly one vertex or not at all, either the interior or the exterior contains no edges from .
For every monotone drawing style and every filled drawing we have
We proceed by induction on the number of vertices in . The desired inequality clearly holds if itself is essentially -connected. Otherwise, let be a simple closed curve with at least one vertex of in its interior and at least one vertex in its exterior such that is either empty or a single vertex. Let and denote the drawings obtained from by removing every edge and vertex of in the exterior of , respectively interior of . Observe that are filled and in , as is monotone. Further observe that , as every edge of lies on one side of .
Now if , then and , since the vertex in is incident to the cell containing curve in both drawings and . Hence, using induction on and we conclude
On the other hand, if , then . Moreover , since the cell of containing curve can have incident vertices only on one side of , as the drawing is filled. Similar as before, we conclude
As suggested by Section 2, we shall now focus on filled drawings that are essentially -connected. Our goal is to determine the parameter . First, we give an exact formula for the number of edges in any filled essentially -connected drawing. The parameter in the following lemma will later be the for the -planar drawings in Section 3. However, we do not require any drawing to be -planar here.
For any , if is a filled, essentially -connected drawing with vertices, then
First observe that, since is filled, no cell has four or more incident vertices. Hence, . By counting along the angles around each vertex, we see that
Note that this relies on the assumption that is essentially -connected, as this guarantees that each non-isolated vertex lies on the boundary of exactly cells.
As is filled, each cell with exactly two vertices on its boundary is incident to either one or two planar edges and each cell with three vertices on its boundary is incident to exactly three planar edges. Moreover, each planar edge is contained in the boundary of exactly two distinct such cells since has no cut-vertices and . By counting along the sides of the planar edges, we see that which together with (1) gives
Consider the planarization of . Since is essentially -connected, has exactly many connected components. Moreover we have
Applying Euler’s formula to we have
Solving for we have:
where both minima are taken over all filled, essentially -connected drawings and can be thought of as an error term for the drawing , which we seek to minimize. Indeed, if is -planar, i.e., each edge is crossed at most times, then . Thus for we have . In the next section we shall see that (in many cases) the minimum is indeed attained by drawings with .
3 Exact Bounds and Saturated Drawings
Recall that we seek to find the sparsest -planar, -saturated drawings in a drawing style that is given by a set of additional restrictions. These -saturated drawings are related to the filled drawings from Section 2.
For any and any , as well as for , every -planar, -saturated drawing is filled.
Consider a -planar, drawing and a cell in with two incident vertices , such that are not connected by an uncrossed edge in the boundary of . We add a new uncrossed edge in that cell, resulting in a new drawing . Clearly, the introduction of did not create any new selfcrossings, incident crossings, multicrossings, or edges being crossed more than times. Hence, for , drawing lies in and was not -saturated.
It remains to consider and rule out that is homotopic to another edge. So let be an edge parallel to which is closest to in the cyclic order of edges incident to . Since incident crossings and selfcrossings are forbidden, and together form a simple closed curve . If is uncrossed, then is not in the boundary of cell . Since incident crossings are forbidden we find edges and connecting to a vertex in the interior and a vertex in the exterior of , respectively. Hence and are not homotopic. On the other hand, suppose that is crossed by some edge . As incident crossings are forbidden, neither nor is an endpoint of . As multicrossings are forbidden, the two endpoints of lie in the exterior and the interior of , respectively. Hence and are not homotopic. ∎
In order to determine the exact edge-counts for min-saturated drawings, we shall find for each drawing style some essentially -connected, -saturated drawings that attain the minimum in Section 2. Motivated by the error term in Section 2, we define tight drawings as those -planar drawings in which 1) every edge is crossed exactly times (so ) and 2) every cell contains exactly one vertex (so ). Observe that tight drawings are indeed -saturated and filled. Note that, to aid readability, isolated vertices are omitted from the drawings in the figures. Namely, the actual drawings have one isolated vertex in each empty cell in the figures. This is also mentioned in the figure captions.
For every and every monotone drawing style of -planar drawings, if is a tight drawing, then .
If is not essentially -connected, then there is a closed curve containing edges of in the interior as well as exterior, such that is either empty or a single vertex. Then the drawing obtained by removing everything inside (and adding an isolated vertex if the resulting cell is empty) is again in by monotonicity and again tight, but has fewer vertices. As is monotone increasing in , it thus suffices to prove the claim for any essentially -connected tight drawing .
Clearly, is filled, as there are no two vertices incident to the same cell. We immediately get , , , , and it follows that . As there is at least one edge in and this is crossed times, Euler’s formula implies that there are at least three cells. Hence and Section 2 gives
[see also Table 1] Let , be a set of restrictions, and be the corresponding drawing style of -planar drawings.
For infinitely many values of , the minimum number of edges in any -vertex -saturated drawing is
|for and .|
|for and .|
|for and .|
|for and .|
|for and .|
|for and .|
We start with the cases when . Here the drawing style is monotone and every -saturated drawing is filled by Section 3. Thus, by Section 3, we have for every tight drawing . This gives the smallest bound when is minimized. In this case is essentially -connected and by Section 2. So it suffices to consider a tight drawing with the smallest possible number of edges.
Next, we shall go through the possible subsets of and determine exactly the value for in two steps.
First, we present a tight (hence filled) drawing with the smallest possible number of edges, which gives by Section 3 the upper bound
In order to verify (4), observe that if , then the lefthand side is at least , while the righthand side is less than . Thus it is enough to verify (4) when . In particular we may assume and for . Similarly, as , we may assume that . Altogether this implies that (4) is fulfilled unless
In summary, for each we shall give a tight drawing with as few edges as possible, and argue that every filled, essentially -connected drawing with fewer edges satisfies the inequality (4). Note that as essentially -connected drawings have at least one edge. In fact, we may assume that contains at least one crossed edge, as otherwise is filled and planar with and thus and . Altogether this already verifies (4) as follows:
- Case 1. and
Figure 2 shows drawings with edge when is even, and edges when
is odd, which are tight forfor both and , as incident edges do not cross. Thus and for even, respectively for odd. Together this gives .
On the other hand, let be any filled, essentially -connected drawing. As argued above, we may assume that . For even , there is nothing to show as . For odd , we may assume that consists of exactly one edge, which has exactly selfcrossings (since ), and some of the resulting cells may contain an isolated vertex. In particular, . Applying Euler’s formula to the planarization of we get , which verifies (4) as follows:
- Case 2. and
Figure 3 shows drawings with edges which are tight for for both and , as there are neither incident crossings nor selfcrossings. Thus and , which gives .
On the other hand, let be any drawing in , and assume again that . In particular, has exactly one crossed edge, which however is impossible as selfcrossings are forbidden.
- Case 3.
Figure 4 shows tight drawings with edges. Thus , which gives
On the other hand, let be any drawing in . As argued above the desired inequality (4) holds, unless and . As there are no multicrossings, the crossed edges may pairwise cross at most once, and additionally each crossed edge may cross itself at most once, i.e., . However, this would imply
and thus . However, then , which contradicts that there are no multicrossings.
- Case 4.
Figure 5 shows tight drawings with edges. Thus , which gives
On the other hand, let be any drawing in . Again (4) holds, unless and . As there are no multicrossings and no selfcrossings, we have . However, this would imply , which is a contradiction.
- Case 5.
For we have , which gives
For we have analogous to Case 3 , which gives
Figure 6: Smallest tight drawings for (left) and (right) in case , i.e. incident crossings and multicrossings are forbidden. (Isolated vertices in empty cells are omitted.)
On the other hand, let be any drawing in . Clearly, for . However, we already argued in Case 3 that there is no drawing in with and . This already seals the deal for .
For , assume that is a filled, essentially -connected drawing in . At least pairs of crossed edges are incident (though not crossing each other). Hence, as multicrossings and incident crossings are forbidden, we have . Now let us consider the subdrawing of the planarization of obtained by removing all vertices of . I.e., is planar, connected, , and . Applying Euler’s formula shows that has exactly faces. As is filled, any face of with vertices of contains at least uncrossed edges of . Thus we conclude:
Hence , as desired, or , which as in Case 3 gives a contradiction with , since multicrossings are forbidden.
- Case 6.