Gap-planar Graphs

08/25/2017 ∙ by Sang Won Bae, et al. ∙ The University of Sydney Tohoku University Universität Passau Technische Universität Darmstadt Università Perugia California State University, Northridge 0

We introduce the family of k-gap-planar graphs for k ≥ 0, i.e., graphs that have a drawing in which each crossing is assigned to one of the two involved edges and each edge is assigned at most k of its crossings. This definition is motivated by applications in edge casing, as a k-gap-planar graph can be drawn crossing-free after introducing at most k local gaps per edge. We present results on the maximum density of k-gap-planar graphs, their relationship to other classes of beyond-planar graphs, characterization of k-gap-planar complete graphs, and the computational complexity of recognizing k-gap-planar graphs.

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1 Introduction

Figure 1: A drawing of a graph (left) and its cased version where each edge is interrupted at most twice, i.e., a -gap-planar drawing of (right).

Minimizing the overall number of edge crossings in a drawing has been the main objective of a large body of literature concerning the design of algorithms to automatically draw a graph. In fact, several graph drawing algorithms assume the input graph to be planar or planarized (that is, crossings are replaced with dummy vertices which are removed in a post-processing step). More recently, cognitive experiments suggested that the absence of specific kinds of edge crossing configurations has a positive impact on the human understanding of a graph drawing [39]. These practical findings motivated a line of research, commonly called beyond planarity, whose focus is on non-planar graphs that can be drawn by locally avoiding specific edge crossing configurations or by guaranteeing specific properties for the edge crossings (see, e.g., [12, 36, 38, 44]).

Among the most investigated families of beyond-planar graphs are: -planar graphs (see, e.g., [13, 42, 46]), which can be drawn with at most crossings per edge; -quasiplanar graphs (see, e.g., [2, 3, 27]), which can be drawn with no pairwise crossing edges; fan-planar graphs (see, e.g., [10, 14, 40]), which can be drawn such that each edge is crossed by a (possibly empty) set of edges that have a common endpoint on one side; RAC graphs (refer, e.g., to [21]), which admit a straight-line drawing with right-angle crossings.

In this paper we introduce a family that generalizes -planar graphs by introducing a nonsymmetric constraint on the intersection pattern of the edges. Intuitively speaking, we charge each crossing to only one of the two edges involved in the crossing and do not allow an edge to be charged many times. This constraint is motivated by edge casing, a method commonly used to alleviate the visual clutter generated by crossing lines in a diagram [5, 26]. In a cased drawing of a graph, each crossing is resolved by locally interrupting one of the two crossing edges; see Figure 1 for an illustration. This edge casing makes only one of the edges involved in the crossing hard to follow whereas the other one is unaffected. Regardless of the number of crossings, the drawing will remain clear as long as no edge is cased many times; thus, an edge could participate in arbitrarily many crossings as long as the other edges are cased. Eppstein et al. [26] studied several optimization problems related to edge casing, assuming the input is a graph together with a fixed drawing. In particular, the problem of minimizing the maximum number of gaps per edge in a drawing can be solved in polynomial time (see also Section 2). We also note that a similar drawing paradigm is used by partial edge drawings (PEDs), in which the central part of each edge is erased, while the two remaining stubs are required to be crossing-free (see, e.g., [17, 18]).

We formalize this idea with the family of -gap-planar graphs, a family of graphs that can be drawn in the plane such that each crossing is assigned to one of the two involved edges and each edge is assigned at most crossings (for some constant ). We present a rich set of results for -gap-planar graphs related to classic research questions, such as bounds on the maximum density, drawability of complete graphs, complexity of the recognition problem, and relationships with other families of beyond-planar graphs. Our results can be summarized as follows:

  • Every -gap-planar graph with vertices has edges (Section 3). If , we prove an upper bound of for the number of edges in a -gap-planar multigraph with vertices (without homotopic parallel edges), and construct -gap-planar (simple) graphs that attain this bound for all . Note that the same density bound is known to be tight for -planar graphs [46].

  • We study relationships between the class of -gap-planar graphs and other classes of beyond-planar graphs. For all , the class of -planar graphs is properly contained in the class of -gap-planar graphs, which in turn is properly contained in the -quasiplanar graphs (Section 4). We note that -planar graphs are known to be -quasiplanar [4, 34]. Furthermore, we investigate the relationship between -gap-planar graphs and -degenerate crossing graphs, a class of graphs recently introduced by Eppstein and Gupta [25].

  • The complete graph is -gap-planar if and only if (Section 5).

  • Deciding whether a graph is -gap-planar is NP-complete, even when the drawing of a given graph is restricted to a fixed rotation system that is part of the input (Section 6). Note that analogous recognition problems for other families of beyond-planar graphs are also NP-hard (see, e.g., [7, 10, 14, 15, 31, 43]), while polynomial algorithms are known in some restricted settings (see, e.g., [6, 10, 15, 20, 24, 37, 35]).

Preliminaries and basic results are in Section 2. Conclusions and open problems are discussed in Section 7.

2 Preliminaries and basic results

A drawing of a graph is a mapping of the vertices of to distinct points, and of the edges of to a continuous arcs connecting their corresponding endpoints such that no edge (arc) passes through any vertex, if two edges have a common interior point in , then they cross transversely at that point, and no three edges cross at the same point. For a subset , the restriction of to the curves representing the edges of is denoted by . A drawing is planar if no two edges cross. A graph is planar if it admits a planar drawing. A planar embedding of a planar graph is an equivalence class of topologically equivalent (i.e., isotopic) planar drawings of . A plane graph is a planar graph with a planar embedding. A planar drawing subdivides the plane into topologically connected regions, called faces. The unbounded region is the outer face.

The crossing number of a graph  is the smallest number of edge crossings over all drawings of . The crossing graph of a drawing is the graph having a vertex for each edge of , and an edge if and only if edges and cross in . The planarization of is the plane graph formed from by inserting a dummy vertex at each crossing, and subdividing both edges with the dummy vertex. To avoid ambiguities, we call real vertices the vertices of that are in (i.e., that are not dummy).

A class of graphs is informally called “beyond-planar” if the graphs in this family admit drawings in which the intersection patterns of the edges are characterized by some forbidden configuration (see, e.g., [36, 38, 44]). Research on such graph classes is attracting increasing attention in graph theory, graph algorithms, graph drawing, and computational geometry, as these graphs represent a natural generalization of planar graphs, and their study can provide significant insights for the design of effective methods to visualize real-world networks. Indeed, the motivation for this line of research stems from both the interest raised by the combinatorial and geometric properties of these graphs, and experiments showing how the absence of particular edge crossing patterns has a positive impact on the readability of a graph drawing [39].

Among the investigated families of beyond-planar graphs are: -planar graphs (see, e.g., [13, 42, 46]), which can be drawn in the plane with at most crossings per edge; -quasiplanar graphs (see, e.g., [2, 3, 27]), which can drawn without pairwise crossing edges; fan-planar graphs (see, e.g., [10, 14, 40]), which can be drawn such that no edge crosses two independent edges; fan-crossing-free graphs [19], which can be drawn such that no edge crosses any two edges that are adjacent to each other; planarly-connected graphs [1], which can be drawn such that each pair of crossing edges is independent and there is a crossing-free edge that connects their endpoints; RAC graphs (refer, e.g., to [21]), which admit a straight-line (or polyline with few bends) drawing where any two crossing edges are perpendicular to each other.

Eppstein et al. [26] studied several optimization problems related to edge casing, assuming the input is a graph together with a fixed drawing. In particular, the problem of minimizing the maximum number of gaps per edge in a drawing can be solved in polynomial time (see also Section 2). We also note that a similar drawing paradigm is used by partial edge drawings (PEDs), in which the central part of each edge is erased, while the two remaining stubs are required to be crossing-free (see, e.g., [17, 18]).

Let be a drawing of a graph . Recall that exactly two edges of cross in one point of , and we say that these two edges are responsible for . A -gap assignment of maps each crossing point of to one of its two responsible edges so that each edge is assigned with at most of its crossings; see, e.g., Fig. 1(right). A gap of an edge is a crossing assigned to it. An edge with at least one gap is gapped, else it is gap-free. A drawing is -gap-planar if it admits a -gap assignment. A graph is -gap-planar if it has a -gap-planar drawing. Note that a graph is planar if and only if it is -gap-planar, and that -gap-planarity is a monotone property: every subgraph of a -gap-planar graph is -gap-planar. The summation of the number of gaps over all edges in a set yields the following.

Property 1.

Let be a -gap-planar drawing of a graph . For every , the subdrawing contains at most crossings.

In fact, the converse of Property 1 also holds, and we obtain the following stronger result.

Theorem 2.

Let be a drawing of a graph . The drawing is -gap-planar if and only if for each edge set the subdrawing contains at most crossings.

Proof.

Property 1 is the only-if direction. It remains to prove the forward direction. Let denote the set of crossings in . Further let denote a set that consists of copies of each edge in . Let be the bipartite graph whose vertex set is and where a crossing is connected to all copies of edges that are responsible for the crossing .

Clearly, -gap-planar assignments correspond bijectively to matchings in such that each crossing is incident to an edge in . By Hall’s theorem, the bipartite graph has a matching from into if and only if for each set , we have .

Let be some subset of crossings. Let denote the set of edges that are responsible for crossings in . By considering the subdrawing , we find . Moreover, by construction of the neighborhood of contains exactly vertices for each edge in , i.e., . Thus it is , which is Hall’s condition. Thus a -gap-planar assignment exists, showing that is -gap-planar. ∎

Note: David Wood (personal communication) has suggested an alternative proof of the above statement: Hakimi [33] proved that a graph has an orientation with maximum outdegree at most if and only if every subgraph has average degree at most . Theorem 2 immediately follows by applying this result to the intersection graph of the edges in a drawing of a graph.

A -gap assignment of a drawing corresponds to orienting the edges of the crossing graph such that each vertex has indegree at most (intuitively, orienting a crossing towards an edge means we assign the crossing to that edge). Since finding an orientation of a graph with the smallest maximum indegree corresponds to finding its pseudoarboricity [28, 47], Property 3 below follows. A pseudoforest is a graph in which every connected component has at most one cycle, and the pseudoarboricity of a graph is the smallest number of pseudoforests needed to cover all its edges.

Property 3.

A graph is -gap-planar if and only if it admits a drawing whose crossing graph has pseudoarboricity at most .

Given a drawing of a graph , we can find the minimum such that is -gap-planar in time, due to the fact that one can find an orientation of with the smallest maximum indegree in time quadratic in the number of edges of  [49].

Note: In an earlier versions of this paper [8, 9] we gave an upper bound on the treewidth of -gap-planar graphs. Our bounds were based on a result by Dujmović, Eppstein, and Wood [23] that bounded the treewidth of a graph as a function of the number of vertices and the crossing number. Unfortunately, their result turned out to be incorrect [xxx]. Thus, it still remains open to show that -gap-planar graphs have treewidth for some function .

3 Density of -gap-planar graphs

We begin with an upper bound on the number of edges of -gap-planar graphs.

Theorem 4.

A -gap-planar graph on vertices has edges.

Proof.

The crossing number of a graph with vertices and edges is bounded by when  [45]. Combined with the bound (Property 1), we obtain

which implies , as required. ∎

Better upper bounds are possible for small values of , in particular for . Pach et al. [45] proved that a graph with vertices satisfies . Combined with the bound , we have

For (i.e., for -gap-planar graphs), this gives . We now show how to improve this bound to (see Theorem 6 below). The idea is to follow a strategy developed by Pach and Tóth [46] and Bekos et al. [13] on the density of 2- and 3-planar graphs, with several important differences.

In order to accommodate the elementary operations in the proof of Theorem 6, we work on a broader class of graphs. A drawing of a multigraph is -gap-planar if it admits a -gap assignment and no two parallel edges are homotopic. A multigraph is -gap-planar if it has a -gap-planar drawing. Two parallel edges are homotopic in a drawing , if the drawings are continuous arcs and , and there is a continuous function such that , , , and for all , and does not map any point of the open square to a vertex in . Intuitively, can be continuously deformed into with fixed endpoints and without passing through any vertex in . In particular, in a -gap-planar drawing, two homotopic parallel edges either cross no other edge (they might cross each other), or they both cross the same edge.

Let , . Let be a -gap-planar multigraph with vertices that has the maximum number of edges possible over all -vertex -gap-planar multigraphs; and let be a -gap-planar drawing of . Let be a sub-multigraph of , where is a maximum multiset of edges that are pairwise noncrossing in , and if there are several such sub-mustligraphs, then has the fewest connected components.

We first show that is a triangulation, that is, a plane multigraph in which every face is bounded by a walk with three vertices and three edges.

Lemma 5.

The multigraph is a triangulation.

The proof of Lemma 5 is deferred to Section 3.1. We can now show that . We state a stronger result (for multigraphs) that immediately implies the same density bound for simple graphs (Corollary 7).

Theorem 6.

A multigraph on vertices that has a -gap-planar drawing in which no two parallel edges are homotopic has at most edges.

Corollary 7.

A -gap-planar (simple) graph on vertices has at most edges.

Proof of Theorem 6..

By Lemma 5, is a triangulation. By Euler’s polyhedron theorem, it has edges and triangular faces. Consider the edges in . It remains to show that .

The embedding of edge is a Jordan arc that visits two or more triangle faces of . We call the first and last triangles along the end triangles of . For an end triangle , the connected component of incident to a vertex of is called an end portion. We use the following charging scheme.

Figure 2: Example for the charging scheme in the proof of Theorem 6. Bold edges are in a crossing-free triangulation . Every edge in is charged to a triangle of as indicated by arrows. Left: a simple graph where edge is charged to , edge to , and edge to . Right: A -gap-planar multigraph with nonhomotopic parallel edges.

Each edge charges one unit to a triangle face of as follows. (Refer to Fig. 2.) If has an end portion that has a gap neither in the interior nor on the boundary of the corresponding end triangle , then charges one unit to . (If neither end portions of has a gap in the interior or on the boundary of its end triangle, then charges one arbitrary end triangle.) Otherwise the two end portions of lie in two adjacent triangles, say, and , and uses its gap to cross an edge on the boundary between and ; in this case charges one unit to or as follows: If has a gap and the edge passing through this gap charges (because has an end portion that has a gap neither in the interior nor on the boundary of ), then charges , otherwise it charges .

We claim that each face of receives at most one unit of charge. Let be a face in . Suppose to the contrary that receives positive charge from two edges, say . Then both edges have an end portion in that do not have gaps in the interior of . Consequently, the end portions of and in cannot cross, and so they are incident to the same vertex of . Therefore, the both end portions and are incident to the same vertex of , say , and cross the edge of opposite to , namely . Let be the face of the plane graph on the opposite side of .

Assume first that the end portion has a gap neither in the interior nor on the boundary of . Then passes through the gap of . Since has at most one gap in a -gap-planar drawing, uses its own gap to cross . By our charging scheme, this implies that , and it must charges one unit to (rather than ). Next assume that does not have any gap. Then and each use their own gaps to cross . Both and are homotopic to an edge lying in by our charging scheme. All cases lead to a contradiction, hence receives at most unit of charge, as claimed. Consequently, is bounded above by the number of faces of , which is , as required. ∎

3.1 Proof of Lemma 5

We start with a few basic observations. Let be an edge-maximal multigraph on vertices that has a -gap-planar drawing without homtopic parallel edges.

Lemma 8.

Graph is connected.

Proof.

Suppose, to the contrary, that is disconnected. Let be one component, and let be the disjoint union of all other components (i.e., and ). For , let (i.e., the drawing of inherited from ), and let be the planarization of .

Let be a face in incident to some vertex . Apply a projective transformation to so that the outer face is incident to some vertex ; followed by an affine transformation that maps into the interior of face . We obtain a -gap-planar drawing of in which we can insert a new crossing-free edge , between two distinct components of , contradicting the maximality of . ∎

Recall that is a -gap-planar drawing of with the minimum number of crossings. We show that this implies that is a simple topological drawing, that is, no edge crosses itself and every pair of edges cross at most once. This follows from standard simplification techniques, but we provide the proof for completeness.

Lemma 9.

is a simple topological drawing.

Proof.

Suppose the drawing of an edge crosses itself at point . Then crosses itself only once, and this crossing is charged to edge , hence all other crossings of are charged to other edges. We can redraw by eliminating the loop of . This yields a new -gap-planar drawing of with at least one fewer crossings, contradicting the minimality of .

Suppose the drawings and of edges and cross at points and . Then they cross exactly twice and these two crossings are charged to and , hence any other crossing of or with some edge is charged to . We can redraw and in by exchanging their subarcs between and such that both crossings are eliminated. This yields a new -gap-planar drawing of with fewer crossings, contradicting the minimality of . ∎

Since is connected, every face in the planarization of has a connected boundary. The boundary walk of a face is a closed walk in such that lies on the left hand side of each edge along the walk; and every two consecutive edges of the walk, and , are also consecutive in the counterclockwise rotation of all edges incident to . Let denote the set of faces in the planarization that are not incident to any vertex in .

Lemma 10.

If , then the boundary walk of is

  1. a simple cycle (i.e., has no repeated vertices) with at least 3 vertices;

  2. disjoint from the boundary walk of any other face in .

Proof.

1. Let , and let be its boundary walk. By Lemma 9, we have . Let be the set of vertices in ; and let be the set of edges in that contain some edge of . It suffices to show that , and then has no repeated vertices, hence it is a simple cycle.

Suppose, to the contrary, that the vertices in are not distinct. Since , all vertices in are crossings in the drawing , consequently they all have degree 4 in the planarization . If , , then and cannot be consecutive vertices in , and two pairs of edges from , , , are part of the same edge in . If , for some , then . This implies . That is, the edges in are involved in more than crossings, contradicting the assumption that is a -gap-planar drawing.

2. Let be two faces, with boundary walks and . Both and are simple cycles by part 1. For , let be the set of vertices in , and the set of edges of that contain the edges of the walk .

Note that and cannot share two consecutive edges, say and , since the middle vertex has degree 4 in . When and have a common edge, say , then three pairs of edges from , , , , are part of the same edge in . When and have a common vertex but no common edge incident to , then two pairs of edges from , , , are part of the same edge in . This implies . That is, the edges in are involved in more than crossings, contradicting the assumption that is -gap-planar. ∎

Figure 3: Illustration for the proof of Lemma 11: Two adjacent faces in the planarization that are incident to two distinct vertices, and , separated by an edge . If we replace the edge by either or both and , we obtain a -gap-planar drawing of a graph that has either one fewer component in or one more edge.

Recall that is a sub-multigraph of , where is a maximum multiset of edges that are pairwise noncrossing in , and if there are several such sub-mustligraphs, then has the fewest connected components.

Lemma 11.

Graph is connected.

Proof.

Suppose, to the contrary, that is disconnected. Let be one component, and let , where and .

Consider the faces in the planarization of . Notice that there is no face in incident to a vertex and a vertex , otherwise we could either add a new edge contradicting the maximality of , or redraw an existing edge to pass through the interior of this face, contradicting the maximality of .

Consequently, we can partition the faces in into three categories: For , let be the set of faces incident to a vertex in ; and let be the set of faces incident to neither nor . By Lemma 10, the region obtained by removing all faces in (i.e., ) is connected. Consequently, there exist some faces and that have a common edge in . Let and be incident to and . Let be the edge on the common boundary of and , and denote its endpoints by .

We consider three possible edges that we describe together with their drawings (up to homotopy equivalence) with respect to : Let be an edge such that it lies in ; let (resp., ) be an edge such that it starts in and closely follows edge from to its endpoint (resp., ). Refer to Fig. 3. (If edge (resp., or ) is homotopic to an existing edge in , then we can redraw it as described above, and maintain a -gap-planar drawing of ).

Note that , otherwise we can add to with the drawing described above, and charge the crossing to , contradicting the maximality of . Note also that and (which may or may not be present in ) form a path between and . We distinguish two cases:

  • Assume . We can add to , contradicting the maximality of .

  • Assume . If neither nor is present in and , then we can modify by replacing with these edges, contradicting the maximality of . If both and are present in , then they both are in by the maximality of . In this case, we can modify by replacing with . Then , , , and will be in the same component of , contradicting the tie-breaking rule that was a maximum crossing-free subgraph with the fewest components. Otherwise we can modify both and by replacing with or (whichever is not already present in ), and then add edge to , which contradicts the maximality of .

All cases lead to a contradiction, which completes the proof. ∎

In the proof of Lemma 5, we shall use Sperner’s Lemma [48], a well-known discrete analogue of Brouwer’s fixed point theorem.

Lemma 12.

(Sperner [48]) Let be a geometric simplicial complex in the plane, where the union of faces is homeomorphic to a disk. Assume that each vertex is assigned a color from the set such that three vertices are colored 1, 2, and 3, respectively, and for any pair , the vertices on the path between and along that does not contain the 3rd vertex are colored with . Then contains a triangle whose vertices have all three different colors.

We are now ready to prove Lemma 5, restated in the following form.

Lemma 13.

The multigraph is a triangulation, that is, a plane multi-graph in which every face is bounded by a walk with three vertices and three edges.

Proof.

Suppose, to the contrary, that is not a triangulation. Then has a face whose boundary walk has more than three vertices (i.e., ). To simplify notation, we assume that is a simple cycle; this assumption is not essential for the proof.

Let be the subgraph of formed by all edges and vertices lying in the interior or on the boundary of ; let denote the set of vertices of (it consists of and all crossings in the interior or on the boundary of ); and let denote the set of faces of that lie in . Let be the set of faces that are not incident to any vertex in ; and for , let be the set of faces incident to .

We note the following properties of the arrangement of faces in .

  • A face cannot be incident to a vertex , . Indeed, otherwise we could add a new edge to that lies in . Note that does not contain a homotopic parallel edge, otherwise it would lie in the face , and could be added to , contradicting the maximality of .

  • A face cannot be adjacent to a face , . Indeed, otherwise we can add a new edge to such that lies in and uses a gap to cross the boundary between these faces (Fig. 4(a)). Again cannot contain a homotopic parallel edge, otherwise it would lie in the face , and could be added to , contradicting the maximality of .

  • A vertex cannot is incident to two faces and , Suppose, to the contrary, that there is such a vertex (Fig. 4(b)). Then two edges cross at . We can replace edge with a new edge that lies in and that crosses edge at . The new edge can be inserted into both and , contradicting the maximality of . In this case, cannot already contain a homotopic parallel edge, otherwise it could be added to , contradicting the maximality of .

  • A face cannot be adjacent to two faces and . Suppose to the contrary that there is such a face (Fig. 4(c)). Then two edges are on the common boundary of the adjacent pairs and . We can replace edge with a new edge that lies in that crosses edge . The new edge can be inserted into both and , contradicting the maximality of . Again, cannot already contain a homotopic parallel edge, otherwise it could be added to , contradicting the maximality of .

We next distinguish two cases.

Case 1. For every , the edge is incident to faces in only. We use Sperner’s Lemma [48] for a triangulation of the dual graph on the faces , that we define here. We first create the standard dual graph of : The nodes correspond to the faces in ; and two nodes are adjacent if and only if the corresponding faces are adjacent in . We then triangulate the standard dual graph as follows. For every crossing in the interior of is incident to four faces in , and their adjacency graph forms a 4-cycle in the standard dual. By Lemma 10(2), at least three of those faces are in . We triangulate the 4-cycle by an arbitrary diagonal between two faces in . Note that the faces in still form an independent set by Lemma 10(2). We call the resulting graph the modified dual graph of . By Property (P4), every face in is adjacent to at most one side of . Consequently, the modified dual graph is 2-connected, and the neighbors each face form a cycle or a path. Finally, remove all nodes corresponding to from the modified dual graph, and triangulate the cycle or path of neighboring nodes arbitrarily to obtain a triangulation . The condition in Case 1 implies that is a geometric simplicial complex, where the union of faces is homeomorphic to a disk.

We now define a 3-coloring of (the coloring need not be proper). Assign color 1 to all faces in . For , assign color 2 to all faces in if is even, and color 3 if

is odd. Since

, each of the three colors are used at least once.

Figure 4: Illustration for the proof of Lemma 13 with : The dual graph (which is not shown in the figure) has a triangle whose nodes have three different colors, say , , and , where . (a) Faces and are adjacent. (b) Vertex is incident to both and . (c) A face is adjacent to both and .

We have seen that satisfies the conditions of Sperner’s Lemma. The Lemma implies that contains a triangle whose nodes have all three different colors, say , , and . Without loss of generality, assume that . (Possibly we have , e.g., , , and .) Consider three cases depending on how the edge in was created:

  • If and are adjacent in , then (P2) is violated.

  • If a vertex is incident to both and , then (P3) is violated.

  • If a face is adjacent to both and , then (P4) is violated.

All three subcases lead to a contradiction.

Case 2. There is an index such that is incident to a face in for some . Without loss of generality, we may assume that edge is incident to a face in for some . (Refer to Fig. 5 where .) Note that edge must be incident to some face in for all ; otherwise would be incident to two faces, and , , that are either adjacent to each other or both adjacent to some face ; and then we could add a new edge lying in or .

It follows that there are faces and that are incident to some point (see Fig. 5(a)); or both are adjacent to some common face that is incident to (see Fig. 5(b)).

Figure 5: Illustration for the proof of Lemma 13 with : Edge is incident to some face in for all . (a) Faces and that are incident to some point . (b) Faces and adjacent to face that is incident to . (c) Face is incident to .

Consider the face of on the opposite side of , and let be the set of faces in the planarization contained in . Let be a face incident to or adjacent to face . By Lemma 10(2), we may assume that is incident to a vertex on the boundary of the face . It is possible that or .

  • If , we modify , , and as follows (Fig. 5(a)–(b)): Consider the possible edges and that lie in and , respectively, they each cross and at most one additional edge at or at a vertex of . If or is present in and (as a homotopic copy), it can be redrawn to lie in and , respectively. If or is not present in and , we then insert it and remove the edge . Finally, we can modify by replacing with and , contradicting the maximality of .

  • If , then we modify , , and as follows (Fig. 5(c)): Add a new edge that lies in or , and crosses at a point on the boundary between and . Then redraw the edges and by exchanging their initial arcs between and , and eliminating the crossing at . The edge was not previously present in , otherwise it would be homotopic to a diagonal of the face of , contradicting the maximality of . (However, a homotopic copy of the new drawing of edge may be already present in , in which case, the total number of edges in remains the same). Modify by replacing the edge of face with the new edges and described here. This contradicts the maximality of .

  • If and , we make similar changes: We increase by replacing the edge of with a new edge and a new drawing of the edge .

All cases lead to a contradiction. Therefore, our initial assumption must be dropped, consequently the multigraph is a triangulation, as claimed. ∎

3.2 Lower bound constructions

We now show that the bound of Theorem 6 is worst-case optimal. A 2-planar graph with vertices and edges is also -gap-planar (see Lemma 19). Pach and Tóth [46] construct such a graph by starting with a plane graph with pentagonal faces (e.g., using nested copies of an icosahedron), and then add all five diagonals in each pentagonal face; see Fig. 6. This construction yields a -gap-planar graph with vertices and edges for all , .

Figure 6: Patterns that produce -gap-planar graphs with vertices and edges.

We can modify this construction by inserting a new vertex in one or more pentagons, and connecting it to the 5 vertices of the pentagon; see Fig. 6. Every new edge crosses exactly one diagonal of the pentagon, so the new crossings can be charged to the new edges. Since every new vertex has degree 5, the equation prevails. By inserting a suitable number of vertices into pentagons, we obtain constructions for such that or . A similar construction is based on hexagonal faces; see Fig. 6. Start with a fullerene, that is, a 3-regular, plane graph with vertices, 12 pentagon faces, and hexagon faces (including the external face). Add diagonals in each face to connect a vertex to their second neighbors (the graph is 2-planar so far); finally insert a new vertex in each face of , and connect them to all vertices of that face. We obtain a -gap-planar graph . The number of vertices is , and the number of edges is . Fullerenes exist for and for all even integers  [16]. This yields a lower bound of for and for all where . However, similarly to the previous construction, the equation prevails if we delete up to 12 vertices inserted into pentagons. Consequently, the upper bound in Theorem 6 is tight for all .

Theorem 14.

For every integer there exists a -gap-planar (simple) graph with vertices and edges.

We mention a third, slightly weaker construction, which is based on a sequence of nested squares. Fig. 6 shows how to add 16 edges between two consecutive squares such that the 16 crossings are assigned to distinct edges. We can add two diagonals in the external face and the innermost square. Using squares, we have , and . In particular, for this yields a drawing of ; see Fig. 7.

If we allow -gap-planar muligraphs (with nonhomotopic parallel edges in a -gap-planar drawing), then we can construct smaller configurations for which the upper bound of Theorem 6 is tight. Start with a regular polygon with vertices. Subdivide the interior and the exterior of independently into one pentagon and triangle faces using diagonals. In each of the two pentagons, add five edges as shown in Fig. 2(left). In each triangle, add a new vertex and six new edges as shown in Fig. 2(right). We obtain a -gap-planar drawing of a multigraph with vertices and edges for all , . By inserting a new vertex in one or two pentagons, and connecting it to the 5 vertices of the pentagon as in Fig. 6, the lower bound extends for all integers . We summarize our lower bound for multigraphs in the following theorem.

Theorem 15.

For every integer , there exists a -gap-planar multigraph with vertices and edges.

4 Relationship between -gap-planar graphs and other families of beyond-planar graphs

In this section we prove the following theorem.

Theorem 16.

For every integer , the following relationships hold.

We begin by showing the following.

Lemma 17.

For all , every -gap-planar drawing is -quasiplanar.

We also need to show that for every there is a -quasiplanar graph that is not -gap-planar. We prove a stronger statement:

Lemma 18.

For all , there is a 3-quasiplanar graph that is not -gap-planar.

Proof.

Let . We construct a graph as follows. Start with and replace each edge by edge-disjoint paths of length . Note that the total number of edges is . Graph is -quasiplanar. Since , it admits a drawing with precisely one crossing. The paths of length 2 can be drawn close to the edges of such that two paths cross if and only if the two corresponding edges of cross. Consequently, admits a drawing in which any two crossing edges are part of two paths that correspond to two crossing edges of , which in turn implies that no three edges in pairwise cross.

Suppose that admits a -gap-planar drawing . Then the total number of crossings is at most . We derive a contradiction by showing that . If we choose one of the paths for each of the edges of independently, then we obtain a subdivision of , therefore there is a crossing between at least one pair of paths. There are ways to choose a path for each of the 9 edges of . Each crossing between two paths in is counted times. Consequently, the total number of crossings in is at least . ∎

We now show that every -planar drawing is -gap-planar. We note that a similar result can be derived from [18] (Lemma 10), but only for the case . A bipartite graph with vertex sets and is denoted as . A matching from into is a set such that every vertex in is incident to exactly one edge in and every vertex in is incident to at most one edge in . The neighborhood of a subset is the set of all vertices in that are adjacent to a vertex in , and is denoted as .

Lemma 19.

For all , every -planar drawing is -gap-planar.

Proof.

let , , let be a -planar graph, and let be a -planar drawing of . Let be a bipartite graph obtained as follows. The set has a vertex for each crossing in between two edges and of . For each edge of there are vertices