(k,p)-Planarity: A Generalization ofHybrid Planarity

-Planarity: A Generalization of
Hybrid Planarity

Emilio Di Giacomo Università degli Studi di Perugia, Italy
11email: alessandra.tappini@studenti.unipg.it {emilio.digiacomo,giuseppe.liotta}@unipg.it
   William J. Lenhart Williams College, US
{wlenhart,twr2}@williams.edu
   Giuseppe Liotta Università degli Studi di Perugia, Italy
11email: alessandra.tappini@studenti.unipg.it {emilio.digiacomo,giuseppe.liotta}@unipg.it
  
Timothy W. Randolph
Williams College, US
{wlenhart,twr2}@williams.edu
   Alessandra Tappini Università degli Studi di Perugia, Italy
11email: alessandra.tappini@studenti.unipg.it {emilio.digiacomo,giuseppe.liotta}@unipg.it
Abstract

A graph is -planar if its vertices can be partitioned into clusters of size at most such that admits a drawing where: (i) Each cluster is associated with a closed region in the plane (called cluster region) that is topologically equivalent to a disk; (ii) Cluster regions are pairwise disjoint; (iii) Each vertex of a cluster is associated with at most distinct points (called ports) on the boundary of its cluster region; (iv) Each edge is a Jordan arc connecting a port of with a port of ; and (v) Edges connecting different clusters do not cross each other and do not intersect any cluster region, with the only exception for end-points. We first ask a Turán-type question and give a tight upper bound on the number of edges of -planar graphs. We then study the complexity of the -planarity testing problem and prove that -planarity testing and -planarity testing are both NP-complete problems. Finally, motivated by the NP-completeness of -planarity testing, we investigate the combinatorial properties of -planar graphs and study their relationship with -planar graphs, describing families of -planar graphs that are -planar as well as families that are not.

1 Introduction

Visualization of non-planar graphs is one of the most studied graph-drawing problems in recent years. In this context, an emerging topic is hybrid representations (see, e.g., [DBLP:conf/gd/AngeliniLBFPR15, Batagelj_Visual_2011, DBLP:conf/gd/LozzoBFP16, dlpt-ntptsc-17, henry2007nodetrix]). A hybrid representation simplifies the visual analysis of a non-planar graph by adopting different visualization paradigms for different portions of the graph. The visualization of some, typically dense, subgraphs called clusters is confined to restricted regions of the plane where the visual complexity is expected to be high because of their many (intra-cluster) edges; the remaining (inter-cluster) edges have low visual complexity. As a result, a hybrid representation provides an easy to read overview of the graph structure and it allows a “drill-down” approach when a more detailed analysis of some of its clusters is needed. The inter-cluster edges are represented as simple Jordan arcs according to the classical node-link graph drawing paradigm, while the clusters and their edges are represented by adopting alternative paradigms. Different representation paradigms for the clusters give rise to different types of hybrid representations. For example, Angelini et al. [DBLP:conf/gd/AngeliniLBFPR15] introduce intersection-link representations, where clusters are represented as intersection graphs of sets of rectangles, while Henry et al. [henry2007nodetrix] introduce NodeTrix representations, where dense subgraphs are represented as adjacency matrices. Hybrid representations have also been used by Batagelj et al. [Batagelj_Visual_2011] who introduce the -clustering model [Batagelj_Visual_2011], where and define the desired topological properties of the clusters and of the graph connecting such clusters, respectively. For instance, in a --clustering of a graph each cluster is a -clique and the graph obtained by contracting each cluster in a single node (called the graph of clusters) is planar. Batagelj et al. [Batagelj_Visual_2011] combine a node-link representation of the graph of clusters (the “ part”) with alternative representations (adjacency matrices, circular layout, etc.) for the clusters (the “ part”). Given a graph and a hybrid representation paradigm , one could ask whether can be represented according to so that there are no crossings among edges that connect different clusters: This problem is called hybrid planarity, and it can be studied assuming either that the clusters are given as a part of the input or that they are not. Da Lozzo et al. [DBLP:conf/gd/LozzoBFP16] prove that the problem of testing whether a clustered graph admits a planar NodeTrix representation is NP-complete, while Di Giacomo et al. [dlpt-ntptsc-17] study NodeTrix planarity when the maximum size of the clusters is bounded by a constant . They prove that if the graph of clusters is a partial -tree, the problem can be solved in time, while for general graphs of clusters the problem is NP-complete for and can be solved in time for . These results apply in the fixed sides scenario (i.e. for each matrix it is specified which edges must be incident on each of the four sides), while in the free sides scenario the problem remains NP-complete for any .

Figure 1: (a) The complete graph . (b) A -planar representation of .
Figure 2: (a) A NodeTrix representation of a -clique and a corresponding representation. (b) An intersection-link representation of a -clique and a corresponding representation.

In this paper, we present a hybrid representation model that generalizes the above described hybrid paradigms. Given a graph and a clustering of its vertices such that each cluster has at most vertices, a representation of is a geometric representation of such that: (i) Each cluster is associated with a closed region, called cluster region, that is topologically equivalent to a disk; (ii) Cluster regions are pairwise disjoint; (iii) Each vertex of the cluster is represented by at most distinct points, called ports, on the boundary of its cluster region; (iv) Each edge with and in different clusters is represented by a Jordan arc connecting a port representing with a port representing . A representation is planar if the curves representing edges that connect different clusters do not cross each other and do not intersect any cluster region, except at end-points. We say that a graph is -planar if it can be clustered so that it admits a planar representation. Figure 1 shows a -planar representation of . Note that in our definition of a representation, we do not specify how to draw the clusters (in particular how to represent the edges between vertices of the same cluster). The idea is that any paradigm that represents each vertex with at most ports can be used to represent the clusters. In this sense, our model generalizes the previous described models. For example, a NodeTrix representation is a representation since for each vertex the are (at most) ports, one on each side of the matrix (see Fig. 2), while the intersection-link representations defined by Angelini et al. [DBLP:conf/gd/AngeliniLBFPR15] are representations for some depending on the specific intersection representation used for clusters. For example, if clusters are represented using axis-aligned unit squares with their upper-left corners along a common line with slope , we obtain a representation (see Fig. 2). The results of this paper are the following:

  • In Section 3, we study the maximum edge density of -planar graphs. We give an upper bound on the maximum number of edges of a -planar graph and prove that this bound is tight for .

  • In Section 4, we study the complexity of recognizing -planar graphs. We prove that -planarity testing and -planarity testing are both NP-complete problems. These results imply that computing the minimum such that a graph is -planar is NP-hard for both and .

  • The NP-completeness of the -planarity testing problem naturally suggests to further investigate the combinatorial properties of -planar graphs. In particular, we ask whether -planar graphs (i.e. graphs that admit a drawing where every edge is crossed at most once) are also -planar. In Section 5, we show that this is not the case, and we give a sufficient condition for -planar graphs to be -planar.

Preliminary definitions are in Section 2, while conclusions and some open problems are discussed in Section 6.

We conclude this section by observing that -planarity can be regarded as a relaxation of clustered planarity [DBLP:conf/compgeom/CorteseB05, cdfpp-cpcpcg-08, ddm-pcg-02, df-ecptefcgsf-09, DBLP:conf/esa/FengCE95, gls-cpecg-06]. Given a graph and a clustering of its vertices, a clustered planar drawing is a planar drawing of such that clusters are represented inside closed regions of the plane and an edge cannot cross twice the boundary of any such regions. The main difference with our model is that -planarity allows multiple ports to be used to represent a given vertex. From this perspective, the use of different ports to represent a vertex can be regarded as an example of vertex splitting [DBLP:conf/gd/EadesM95, Eppstein2018, FARIA200165, Hartsfield1985, Jackson1984]; however, while in the papers that use vertex splitting to remove crossings the multiple copies of each vertex can be placed anywhere in the drawing, in our model they are forced to lay within the boundary of the same cluster region.

2 Preliminaries

Let be a graph whose vertices are partitioned into disjoint sets called clusters. An edge connecting vertices that belong to the same cluster is called intra-cluster, while an edge connecting vertices that belong to different clusters is called inter-cluster. We say that a representation of is a representation if: (i) is -clustered, i.e. its clusters have size at most ; (ii) each cluster is associated with a distinct closed region, called a cluster region, that is homeomorphic to a disk; (iii) cluster regions are pairwise disjoint; (iv) each vertex is associated with at most distinct points, called ports, on the boundary of its cluster region; and (v) every inter-cluster edge is a Jordan arc connecting a port of with a port of . We say that a graph is -planar if it can be -clustered such that it admits a planar representation, i.e. a representation where inter-cluster edges do not cross each other and do not intersect any cluster region, with the only exception for end-points.

By definition -planar graphs and -planar graphs include -planar graphs for all . Moreover, for any number of ports, the class of -planar graphs is equivalent to the class of planar graphs. Indeed, in every representation of a -planar graph , each cluster is associated with a cluster region whose boundary has at most occurrences of the same vertex, thus each cluster region of can be collapsed into a single vertex to obtain an equivalent representation.

A -planar graph is one that has an embedding in the plane, called -planar embedding, where each edge is crossed at most once. A -planar graph together with a -planar embedding is called a -plane graph. It may be worth remarking that -planar graphs are among the most studied families of graphs in the context of graph drawing beyond planarity (see, e.g., [DBLP:journals/dam/BachmaierBHNR17, DBLP:journals/tcs/BrandenburgDEKL16, didimo20161, DBLP:journals/corr/abs-1804-07257, DBLP:journals/csr/KobourovLM17]).

3 Edge Density of -Planar Graphs

In this section, we give a tight bound on the number of edges of a -planar graph. We start with some additional definitions. A skeleton of a -planar representation is a planar drawing obtained by the following transformation. We first replace each port in with a vertex. Each cluster region of is now an empty convex space surrounded by up to vertices. We connect these vertices in a cycle and triangulate the interior. For our purposes any triangulation is equivalent. The resulting representation is . Figure 3 illustrates a skeleton of the -planar representation of Fig. 3.

Figure 3: (a) A -planar representation of a graph ; (b) A skeleton of .
Theorem 3.1

Let be a -planar graph with vertices. has edges. Also, this bound is tight for any positive integers , and , such that , , and , where .

Proof

Let be a -planar representation of and let be the number of clusters of . Since each cluster can have at most intra-cluster edges, has at most intra-cluster edges. We now show that has at most inter-cluster edges. Let be a skeleton of . Denote by and the number of vertices and the number of edges of . The value is equal to the number of inter-cluster edges plus the number of edges added in place of each cluster region of to create . If a cluster region has ports, connecting the ports with a cycle and triangulating the interior cycle creates additional edges if , or additional edges if . Letting be the number of clusters of consisting of a single vertex, so that , we have that or, regrouping, . Note that , hence . Because is planar, we have and therefore

(1)

Since , we obtain .

To obtain the desired bound, we express the value of in terms of and . If all clusters have size , then and therefore . If the clusters have size at most , denote by the clusters of and let be the size of cluster . We first add non-crossing inter-cluster edges so that the faces of external to the cluster regions are triangles. Let be the resulting -planar representation. Notice that can have multiple edges. We then construct a sequence of -planar representations so that has all clusters of size and each is obtained from by taking into account the cluster . We denote by and the number of vertices and edges of , respectively. If , cluster is not modified and we set . If , we remove the single vertex in and we triangulate the face that is created by this removal (see Fig. 4 and Fig.4). Also in this case multiple edges can be introduced. The number of vertices of is then , while the number of edges is . If , we augment the cluster with dummy vertices, we add ports in between two consecutive ports associated with two different vertices of (see Fig. 4 and Fig. 4). We then add edges internally to and edges externally to to triangulate the face enlarged by the insertions (again multiple edges can be created). The number of vertices of is , while the number of edges of is . We now prove the following claim that together with the fact that (because has all clusters of size ) implies that . Since and , the statement follows.

Figure 4: (a) A cluster of size , corresponding to a vertex ; (b) Removal of and triangulation; (c) A cluster of size ; (d) Augmentation of with ports and triangulation of the face enlarged by the insertions.
Claim 1

If then .

Clearly, nothing has to be proven for . If , we have which gives . In order to prove that Claim 1 holds in this case, we show that , which can be rewritten as . Since we have , which is greater than or equal to for any integer value of . Consider now the case ; notice that this case is possible only for . We have , which in turn gives . Again, we prove that . Rearranging, we obtain ; since , we have , which holds for every , hence the claim follows.

In order to show that the bound is tight for and , we describe a -planar representation with clusters and inter-cluster edges. is possible for any pair of positive integers and such that and for any . has clusters each with vertices and thus ports. Let and be two cluster regions. We say that and are kp-connected if they are connected by edges as shown in Fig. 5(a) (note that, since the number of inter-cluster edges between two -clusters is at most , we can create edges between and only if ). More precisely, , which we refer to as the small end of the -connection, is connected by means of consecutive ports; the first ports have incident edges each, and the last port has an additional edge. , which we refer to as the large end of the -connection, is connected by means of consecutive ports, each connected to one or two edges. Notice that, since we use ports for the large end, for the small end and two ports can be shared by the two ends, each cluster region can be the small end of one -connection and the large end of another -connection. Thus, we can create a cycle with clusters as shown in Fig. 5(b). In the resulting representation there are two faces of degree : One is the outer face and the other one is inside the cycle. By triangulating these two faces with edges for each face, we obtain the -representation . The number of inter-cluster edges of is , while the number of intra-cluster edges is (each cluster has size ). Thus, .

Figure 5: (a) A -connection of two cluster regions and (, ). (b) A cycle of clusters; the bold edges highlight the two faces of degree .

4 Recognition of -Planar Graphs

In this section, we study the -planarity testing problem. It is not difficult to see that -planarity for is equivalent to standard planarity, while for it is equivalent to IC-planarity, whose test is known to be NP-complete [DBLP:journals/tcs/BrandenburgDEKL16]. We recall that IC-planar graphs are -planar graphs with a -planar embedding such that no two pairs of crossing edges share a vertex [Zhang2013]. As a consequence of the NP-completeness of -planarity, the problem of computing the minimum such that a graph is -planar for is NP-hard.

Theorem 4.1

-planarity testing can be performed in linear time for , and it is NP-complete for .

Proof

Let be a representation of a -clustered graph for . Replace each cluster of of size with an -clique. Since the obtained drawing is planar. Thus, -planar graphs with coincide with planar graphs, which can be recognized in linear time.

A graph is -planar if and only if it is IC-planar. If is IC-planar, consider an IC-planar embedding of (i.e., a -planar embedding of such that no two pairs of crossing edges share a vertex) and replace each pair of crossing edges of with a -cycle whose vertices are exactly the vertices of and , so to obtain a -planar representation of .

If has a -planar representation, each cluster of is a subgraph of a -clique and therefore it can be drawn inside its cluster region with at most one crossing, thus obtaining an IC-planar embedding of .

Since testing IC-planarity is NP-complete [DBLP:journals/tcs/BrandenburgDEKL16], -planarity testing is also NP-complete.

Corollary 1

The problem of computing the minimum value of such that a graph is -planar is NP-hard.

We now focus on the -planarity testing problem, that we call -Planarity. We show that -Planarity is NP-complete by a reduction from Planar Monotone 3-SAT [DBLP:journals/ijcga/BergK12]. Also in this case, a consequence of this result is that minimizing such that a graph is -planar is NP-hard. An instance of 3-SAT is monotone if every clause consists solely of positive literals (positive clause) or solely of negative literals (negative clause). A rectilinear representation of a 3-SAT instance is a planar drawing where each variable and clause is represented by a rectangle, all the variable rectangles are drawn along a horizontal line, and vertical segments connect clauses with their constituent variables. A rectilinear representation is monotone if it corresponds to a monotone instance of planar 3-SAT where positive clauses are drawn above the variables and negative clauses are drawn below the variables, as shown in Fig. 6. Given a monotone rectilinear representation corresponding to a boolean formula , the problem Planar Monotone 3-SAT asks if has a satisfying assignment.

Figure 6: (a) A planar monotone representation of . (b) A -planar representation of a -vertex and its associated subgraph.

We denote as the graph created by removing two adjacent edges from the complete graph . In our reduction we make use of the following transformation. Let be a vertex of , we replace with a copy of by identifying with the vertex of with degree . After performing this operation we say that is a -vertex. The following lemma states a useful property of the -vertices.

Lemma 1

Let be a -vertex of a graph and let be the subgraph associated with . In any -planar representation of , is partitioned into clusters of size .

Proof

Suppose that there exists a -planar representation of a -vertex that leaves unclustered or clustered with a vertex not belonging to the same subgraph. If the remaining vertices of the subgraph are grouped into at least five clusters, it does not admit a -planar representation because its graph of clusters includes a subgraph.

Alternatively, suppose that the remaining vertices of the subgraph are grouped into four clusters, in which case the subgraph consists of three -clusters and two -clusters. Each -cluster can contain at most intra-cluster edge, so any -planar representation of this clustering has inter-cluster edges. However, by Equation 1, we have that in any -planar representation of a graph , where is the number of clusters consisting of a single vertex and is the total number of vertices in the skeleton of . This implies that , a contradiction. Thus any -planar representation of a -vertex has only clusters of size , as shown in Fig. 6.

Figure 7: (a) The variable cycle of and false literal boundaries. (b) A clause gadget . (c) A -planar representation of the clause gadget .

Given an instance of Planar Monotone 3-SAT, we construct a graph that is a Yes instance of -Planarity if and only if is a Yes instance of Planar Monotone 3-SAT. For convenience, figures show the construction of the graph corresponding to the Planar Monotone 3-SAT instance in Fig. 6. We represent -vertices and their associated subgraphs with solid dots, while ordinary vertices are represented with hollow dots.

For each variable of (with ), create in a -vertex and connect such -vertices in a cycle, in the order implied by (refer to Fig. 7). Split each edge (, ) of the cycle with a -vertex . Split the edge (, ) with the vertices and . Finally, duplicate the edge and split the duplicated edges with the -vertices and . We refer to this subgraph as the variable cycle. Given a variable , let be the number of positive clauses and be the number of negative clauses of in which appears. For , connect to with a path of ordinary vertices of length equal to . We refer to these paths as false literal boundaries.

For each clause in , create a corresponding clause gadget in . Create ordinary vertices , , and , create a -vertex , and add an edge between any pair of vertices, as in Fig. 7. Observe that in any -planar representation of a clause gadget, two of the four vertices and must be arranged in one cluster of size . Indeed, by Lemma 1, must be clustered within its subgraph. If and were all clustered separately, the graph of clusters of would contain a minor. Also, any -clustering of a clause gadget in which a literal vertex is clustered with is -planar, as shown in Fig. 7.

Figure 8: (a) The graph . (b) A drawing of the variable cycle of with false literal boundaries oriented according to variable assignment.

Now, connect the clause gadgets with a tree structure corresponding to the positions of clause rectangles in . Let be a clause rectangle in with , , and corresponding to the vertical segments descending from from left to right. If is nested between vertical segments corresponding to literals and of another clause rectangle , split the edges and with K-vertices and connect the new K-vertices with an edge. If is nested under no other clause rectangle, split with a K-vertex and connect the new vertex to if corresponds to a positive clause and to otherwise. This procedure leads to a configuration consisting of two trees of clause gadgets connected as in Fig. 8. This concludes the construction of , which is -planar if and only if has a satisfying assignment.

Theorem 4.2

-Planarity is NP-complete.

Proof

-Planarity is trivially in NP. We now prove the NP-hardness of -Planarity. Let be a Yes instance of Planar Monotone 3-SAT, and let be an assignment function satisfying . We show that the graph corresponding to is -planar by constructing a -planar drawing of using as a template, as explained in the following.

Replace each variable rectangle in with the corresponding vertex of and draw the variable cycle. We refer to the region defined by the variable cycle and the () vertex as the positive side (negative side). For each variable , draw its false literal boundary on the negative side if and on the positive side if . Fig. 8 illustrates a drawing of the variable cycle and false literal boundaries of according to the assignment of and to and and to .

Let be the literal vertex corresponding to clause and variable . Place at the point of intersection between the rectangle associated with and the vertical segment connecting the rectangles and .

Connect the three literal vertices of to form a face, and insert and on the interior, creating one necessary crossing. Insert the tree structure edges, which by construction can be added without creating crossings. Connect literal vertices to variable vertices, which creates a crossing on a false literal boundary precisely when the value assigned to a variable by does not match the literal. Fig. 9 illustrates such a drawing of .

Resolve each crossing at a false literal boundary by clustering the literal vertex with a vertex on the boundary. The specification that each false literal boundary has at least vertices ensures that this operation can be performed. Because satisfies , each clause gadget has at least one literal vertex that can be connected to its variable vertex without crossing a false literal boundary. Cluster this vertex with to resolve each clause gadget crossing. The result of this process is a -planar drawing of as illustrated in Fig. 9.

Figure 9: (a) A drawing of the graph . (b) A -planar representation of .

Let be a Yes instance of -Planarity corresponding to an instance of Planar Monotone 3-SAT. We show that is a Yes instance of Planar Monotone 3-SAT.

Let be a -planar representation of . First, note that any vertices and connected by an edge in must be drawn on the same side of the variable cycle in any -planar representation of . This follows from Lemma 1, as neither nor can be clustered with any K-vertex in the variable cycle. Thus the positive (negative) clause gadgets must all be drawn on the same side of the variable cycle as they are connected by the tree structure to () and the variable vertices. We refer to the sides of the cycle with the positive and negative clause gadgets as the and sides of the cycle. As a consequence of Lemma 1, each false literal boundary is drawn either on the positive or on the negative side of the cycle as well.

Define an assignment function by setting to () if the false literal boundary for is drawn on the () side of the vertex cycle in . We claim that at least one literal vertex of each positive (negative) clause gadget is connected in to a variable vertex with set to True (False).

Without loss of generality, consider the case of a positive clause gadget with literals , , and connected to variables , , and . Assume for contradiction that every literal vertex of is connected in to a variable with , which means that the false literal boundaries of , , and are drawn on the positive side of the variable cycle. We show that any placement of the K-vertex creates an edge crossing in , contradicting our assumption.

Figure 10: Possible placements of the clause vertex relative to its three corresponding clause boundaries.

Suppose first that is placed outside the false literal boundaries. Then each of , , and must be clustered with a boundary vertex and the clause gadget does not admit a -planar drawing (see Fig. 10). Now suppose that is drawn inside the false literal boundary of one constituent variable, for example. In this case, the path (, , ) intersects two false literal boundaries. Because and are -vertices, only can be clustered with a false literal boundary vertex and thus this placement creates at least one necessary crossing (see Fig. 10). Likewise, suppose that is drawn inside the false literal boundary of two constituent variables, for example, and . In this case, the path (, , ) crosses three false literal boundaries and creates a necessary crossing (see Fig. 10). Finally, suppose that is drawn inside all three false literal boundaries (see Fig. 10). In this case, the path (, , ) crosses two false literal boundaries and creates a necessary crossing. Thus, regardless of the position of the a vertex in , at least one of the literal vertices of must match the assignment of its associated variable vertex. This concludes the proof of our claim, i.e., that at least one literal vertex of each clause gadget in is connected to a variable with . Thus is a satisfying assignment for , and is a Yes instance of Planar Monotone 3-SAT.

Corollary 2

The problem of computing the minimum value of such that a graph is -planar is NP-hard.

5 -Planarity and -Planarity

The NP-completeness of the -Planarity testing problem stated by Theorem 4.2, naturally suggests to further investigate combinatorial properties of -planar graphs. We concentrate on studying the relationship between -planarity and -planarity. This is partly motivated by the interest into -planar graphs in graph drawing (see, e.g., [DBLP:journals/csr/KobourovLM17]) and partly by the following observation. Since a -planar graph admits a drawing where each edge is crossed by at most one other edge, it seems reasonable to remove each crossing of the drawing by clustering two of the vertices that are involved in the crossing and by considering two copies for each of them. An example of this intuition is given in Fig. 11. Also, recall that an -vertex -planar graph has at most edges [DBLP:journals/combinatorica/PachT97] and that, by Theorem 3.1, a -planar graph with vertices has at most edges, which does not immediately imply that there are -planar graphs that are not -planar because of their edge density.

Figure 11: Removal of a crossing through a representation.

As we are going to show, however, there is an infinite family of -planar graphs are not -planar for any value of . On the positive side, we shall describe a non-trivial subset of -planar graphs that are -planar.

Theorem 5.1

For every , there exists a -planar graph with vertices and edges that is not -planar, for any .

Proof

We define a recursive family of -plane graphs as follows. Graph consists of a single kite , which is a -plane graph isomorphic to drawn so that all the vertices are on the boundary of the outer face. Graph , for , has kites in addition to ; these kites form a cycle in the outer face of , and each kite contains a vertex of the boundary of the outer face of (note that has vertices on the boundary of the outer face). See Fig. 12 for an example. The kites of are called the external kites of . The embedding of described in the definition will be called the canonical embedding of . We also consider another possible embedding, called the reversed embedding. Let be the boundary of the outer face in the canonical embedding of ; in the reversed embedding of the cycle is the boundary of an inner face and all the rest of the graph is embedded outside . See Fig. 12 and Fig. 12 for an example. For any , let be a copy of with a canonical embedding, and let be a copy of with a reversed embedding. The graph obtained by identifying the external kites of with the external kites of is denoted as . Fig. 12 shows the graph . By construction, has vertices and edges. Hence, has vertices and edges.

We show that is not -planar for any . Suppose that has a -planar representation for some and let be the graph of clusters of . Since is planar, must be planar. can be obtained from by contracting each pair of vertices that is assigned to each cluster region (and removing multiple edges). Contracting a pair of vertices and , the number of vertices reduces by one and the number of edges reduces by the number of paths of length at most connecting and (for each path we remove one edge). In , there are at most such paths between any pair of vertices. Hence, if we contract pairs of vertices, the number of vertices in is , while the number of edges is . If is planar, and thus it must be , which gives , i.e. we must contract at least pairs of vertices. Since there are vertices, we can contract at most pairs. Thus, it must be , i.e., , which can be satisfied only for .

Note that our argument is independent of the -planar embedding of . This implies that the result holds for -planar graphs, not just for -plane graphs.

Figure 12: (a) Definition of . (b)–(c) Canonical and reversed embedding of . (d) .

Theorem 5.1 motivates further investigation of the relationship between -planar and -planar graphs. Note that there are infinitely many -planar graphs that are not -planar. For example, every graph obtained by connecting with an edge a planar graph and has such a property because is not -planar (it has more than edges) but it is -planar, as depicted in Fig. 1. In what follows, we describe a non-trivial family of -planar graphs that are also -planar.

Let be a -plane graph, and let and be a pair of crossing edges of . Any pair , with , is a representative pair of the edge crossing defined by . An independent set of distinct representatives (ISDR for short) of is a set of representative pairs such that there is exactly one representative pair per crossing and no two representative pairs in the set have a common vertex. For example, Fig. 13 shows an ISDR for the graph of Fig. 13. We want to show that if a -plane graph has an ISDR then it is -planar. The crossing edges graph of , called -graph for short and denoted as , is the subgraph of induced by the crossing pairs of . is pseudoforestal if is a pseudoforest (i.e. it has at most one cycle in each connected component). For example, the -planar graph of Fig. 13 is pseudoforestal, as shown in Fig. 13.

Figure 13: (a) A -planar graph . (b) An ISDR of . For each pair of crossing edges the representative pair is indicated with a dashed line connecting the pair. Vertices shared by different crossing pairs are replicated in each pair. (c) The -graph of . The highlighted component contains one cycle, while all others components are acyclic.
Theorem 5.2

A pseudoforestal -plane graph is -planar.

Proof

We start by proving that a -plane graph contains an ISDR if and only if is pseudoforestal. It is known that a graph can be oriented such that the maximum in-degree is if and only if its pseudoarboricity is (i.e. the edges of can be partitioned into pseudoforests) [DBLP:conf/isaac/Kowalik06]. Thus, is pseudoforestal if and only if can be oriented so that the maximum in-degree is one. We now show that this is a necessary and sufficient condition for the existence of an ISDR in . Assume that an ISDR exists. Let and be two crossing edges and let () be the representative pair of and . Direct towards and towards . Doing this for each pair of crossing edges defines an orientation for all edges of . In this orientation each vertex of has in-degree at most , since no two pairs in share a vertex. Now suppose that has an orientation such that each vertex has in-degree at most . For each pair of directed crossing edges in , we add the pair to . Since each vertex in has in-degree at most , is a vertex of at most one pair in . Thus, the pairs selected for different crossing pairs are distinct and no two of them share a vertex.

Figure 14: (a) Two crossing edges and ; (b) Construction of the cluster region and replacement of and ; (c) The resulting drawing.

We now describe how to use an ISDR of to construct a -planar representation of where each pair in is represented as a -cluster that has copies for each of its vertices. Let be a -planar drawing of that respects the -planar embedding of . Consider any two crossing edges and and denote by the point where they cross in . Without loss of generality, assume that is the representative pair of and (see Fig. 14 for an illustration). Subdivide the edge with a copy of placed between and along ; analogously, subdivide the edge with a copy of . Add a curve connecting to and a curve connecting to . By walking very close to the two edges and , these two curves can be drawn without crossing any existing edge and so that the closed curve formed by and together with the portion of from to and the portion of from to does not contain any vertex of . Curve defines the cluster region for the cluster containing and . Replace the edge with a curve connecting to and the edge with a curve connecting to . Again, by walking very close to the two edges and , and can be drawn without crossing existing edges and without crossing each other. The replacements of with and of with remove the crossing between and . Repeating the described procedure for every pair of crossing edges, all crossings are removed. Since for each pair of crossing edges there is a distinct representative pair and no two pair share a vertex, the result is a -planar representation of .

We remark that the pseudoforestal -planar graphs include non-trivial subfamilies of -planar graphs, such as IC-planar graphs (their -graph has maximum vertex degree one) or the -planar graphs such that each vertex is shared by at most two crossing pairs (their -graphs has maximum vertex degree two).

6 Conclusions and Open Problems

In this paper we introduced and studied -planar graphs. We have proven an upper bound on the number of edges of a -planar graph and we have shown that this bound is tight for . This result naturally motivates the problem of establishing a tight bound for . We proved that -planarity testing is NP-complete for and , and for and . It would be interesting to study the complexity of the problem for values of and different from those of Section 4. Finally, we have investigated the relationship between the family of -planar graphs and -planar graphs. We have shown that none of the two families is included in the other and we have described a subfamily that belongs to their intersection. An interesting research direction is to further investigate the relationship between -planar graphs and -planar graphs. For example, one could investigate the relationship between -planar graphs and optimal -planar graphs (i.e. -planar graphs that have exactly edges). More in general, one could compare -planar graphs and -planar graphs for values of larger than .

Acknowledgements

The authors wish to thank Maurizio Patrignani for useful discussions.

References

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