1 Introduction
A flat clustered graph consists of a graph and a set of vertex disjoint subgraphs of called clusters. An edge connecting two vertices in different clusters is an intercluster edge while an edge with both endvertices in a same cluster is an intracluster edge. A hybrid representation of is a drawing of the graph that adopts different visualization paradigms to represent the clusters and to represent the intercluster edges. For example, Fig. 1(a) depicts a flat clustered graph and Fig. 1(b) shows a NodeTrix representation of this graph.
A NodeTrix representation is a hybrid representation of a flat clustered graph where the clusters are depicted as adjacency matrices and the intercluster edges are drawn according to the nodelink paradigm. NodeTrix representations have been introduced to visually explore nonplanar networks by Henry et al. [11] in one of the most cited papers of the InfoVis conference [8]. They have been intensively studied in the last few years, see e.g. [4, 5, 9, 10].
PolyLink representations are a generalization of NodeTrix representations. In a PolyLink representation every vertex of each cluster has two copies that lie on opposite sides of a convex polygon (in a NodeTrix representation the polygon is a square). For example, Fig. 1(c) shows a planar PolyLink representation of the graph of Fig. 1(a).
Intersectionlink representations are another example of hybrid representations: Each vertex of is a simple polygon and two polygons overlap if and only if there is an intracluster edge connecting them [2, 3]. Fig. 1(d) is an intersectionlink representation with unit squares.
Given a flat clustered graph and a hybrid representation paradigm, it makes sense to ask whether is hybrid planar, that is, whether admits a drawing in the given paradigm such that no two intercluster edges cross. In general terms, hybrid planarity testing is a more challenging problem than “traditional” planarity testing. Hybrid representations allow more than one copy for each vertex, which facilitates the task of avoiding edge crossings but makes the problem of testing the graph for planarity combinatorially more complex.
Hybrid planarity testing can be studied both in the “fixed sides scenario” and in the “free sides scenario”. Let be an intercluster edge where is a vertex of cluster and is a vertex of cluster . The fixed sides scenario specifies the sides of the geometric objects representing and to which edge is incident; the free sides scenario allows the algorithm to choose the sides of incidence of edge . Note that what makes the problem challenging even in the fixed sides scenario is that one can permute the different copies of the vertices along the side to which is incident. For example, in the NodeTrix Planarity testing with fixed sides it is specified whether is incident to the top, bottom, left, or right copy of in the matrix representing , and to the top, bottom, left, or right copy of in the matrix representing .
This paper studies different variants of hybrid planarity testing in the fixed sides scenario. It adopts a unified approach that models these problems as instances of a suitably defined constrained planarity testing problem on a graph . The constrained planarity problem specifies for each vertex which cyclic orders for the edges of incident to are allowed. Choosing an order for a vertex of influences the allowed orders for other vertices of ; such dependencies between different allowed orders are expressed by a directed acyclic graph (DAG) whose nodes are FPQtrees (a variant of PQtrees). Our contribution is as follows:

We introduce and study Fixed Constrained Planarity and show that this problem can be solved in quadratic time for biconnected graphs, by modeling it as an instance of Simultaneous FPQOrdering. Fixed Constrained Planarity generalizes the partially PQconstrained planarity testing problem studied by Bläsius and Rutter [6]. Our solution exploits a new definition of fixedness that simplifies and extends results of [6].

We show that a relaxation of NodeTrix Planarity with fixed sides, that allows the rows and the columns of the matrices to be independently permuted, can be modeled as an instance of Fixed Constrained Planarity, and hence it can be solved in quadratic time if the multigraph obtained by collapsing the clusters to single vertices is biconnected. It may be worth recalling that NodeTrix Planarity with fixed sides is NPcomplete in general, but it is lineartime solvable if the rows and the columns of each matrix cannot be permuted [9]. Therefore it makes sense to further explore the conditions under which the problem is polynomially tractable.

We introduce PolyLink representations and we show that biconnected instances of PolyLink Planarity with fixed sides can be solved in quadratic time. As a byproduct, we obtain that a special instance of intersectionlink planarity, called clique planarity with fixed sides, can be solved in quadratic time. Note that clique planarity is known to be NPcomplete in general [2]. We remark that PolyLink Planarity becomes equivalent to NodeTrix Planarity if the polygons have maximum size four and each side is associated with the same set of vertices.
The rest of the paper is organized as follows. Section 2 reports preliminary definitions. In Section 3 we study the Fixed Constrained Planarity testing problem and show that it can be solved in quadratic time for biconnected graphs. In Section 4 we model the RowColumn Independent NodeTrix Planarity testing problem as an instance of Fixed Constrained Planarity; we also introduce the notion of PolyLink representations and we show their relation with other hybrid representation paradigms.
2 Preliminaries
PQtrees: A PQtree is a data structure that represents a family of permutations on a set of elements [7]. In a PQtree, each element is represented by a leaf node, and each nonleaf node is either a Pnode or a Qnode. The children of a Pnode can be arbitrarily permuted, while the order of the children of a Qnode is fixed up to a reversal. Three main operations are defined on PQtrees [6, 7]. Let be a PQtree and let be the set of its leaves. Given , the projection of to , denoted as , is a PQtree that represents the orders of allowed by , such that contains only the leaves of that belong to . is obtained form by removing all leaves not in and simplifying the result, where simplifying means, that former inner nodes now having degree 1 are removed iteratively and that degree2 nodes together with both incident edges are iteratively replaced by single edges. The reduction of with , denoted as , is a PQtree that represents only the orders represented by where the leaves of are consecutive. A Qnode in can determine the orientation of several Qnodes of , while if we consider a Pnode in , there is exactly one Pnode in that depends on . We say that stems from . Given two PQtrees and , the intersection of and , denoted as , is a PQtree representing the orders of represented by both and . If and have the same leaves, their intersection is obtained by applying to a sequence of reductions with subsets of leaves whose orders are given by [6].
Simultaneous PQOrdering: An instance of Simultaneous PQOrdering [6] is a DAG of PQtrees that establishes relations between each parent node and its children nodes. Informally, the DAG imposes that the order of the leaves of a parent node must be “in accordance with” the order of the leaves of its children.
More formally, let be a set of PQtrees whose leaves are , respectively.
Let be a DAG with vertex set and such that every arc in is a triple where is the tail vertex, is the head vertex, and is an injective mapping from the leaves of to the leaves of (). Given two cyclic orders and defined by and , respectively, we say that extends if is a suborder of .
The Simultaneous PQOrdering problem asks whether there exist cyclic orders of , respectively, such that for each arc , extends .
Let be an arc in . An internal node of is fixed by an internal node of (and fixes in ) if there exist leaves and such that (i) removing from makes , , and pairwise disconnected in , and (ii) removing from makes , , and pairwise disconnected in .
An instance of Simultaneous PQOrdering is normalized if, for each arc and for each internal node , tree contains exactly one node that is fixed by .
Every instance of Simultaneous PQOrdering can be normalized by means of an operation called the normalization [1, 6], which is defined as follows. Consider each arc and replace with in , that is, replace tree with its intersection with the projection of its parent to the set of leaves of obtained by applying mapping to the leaves of .
Consider a normalized instance . Let be a Pnode of a PQtree with parents and let be the unique node in , with , fixed by . The fixedness of is defined as , where is the number of children of
containing a node that fixes .
A Pnode is fixed if . Also, instance is fixed if all the Pnodes of any PQtree are fixed.
FPQtrees: An FPQtree is a PQtree where, for some of the Qnodes, the reversal of the permutation described by their children is not allowed. To distinguish these Qnodes from the regular Qnodes, we call them Fnodes [12]. The study of Bläsius and Rutter on Simultaneous PQOrdering also considers the case in which the permutations described by some of the Qnodes are totally fixed, hence the results given in [6] for Simultaneous PQOrdering also hold when the nodes of the input DAG are FPQtrees. In the rest of the paper we talk about Simultaneous FPQOrdering to emphasize the presence of Fnodes, since they play an important role in our applications of hybrid planarity testing.
Embedding DAG: Let be a biconnected planar graph and let be an SPQRdecomposition tree of . The embedding DAG of , denoted as , describes for each vertex , the cyclic orders in which its incident edges appear in any planar embedding of . These cyclic orders can be described by looking at the SPQRdecomposition tree of . We can “translate” an SPQRdecomposition tree of into a set of PQtrees (the embedding trees), which represent the same combinatorial embeddings as the ones defined by [6, Section ]. Note that the cyclic orders around a vertex depend in general on the cyclic orders of the edges around other vertices.
Bläsius and Rutter describe how to express such dependencies and all the planar embeddings of a graph into a DAG of PQtrees by describing and exploiting the relation between PQtrees and SPQRtrees [6, Section ]. The obtained DAG of PQtrees is the embedding DAG , whose size is linear in the size of the SPQRtree , and thus linear in the size of itself. has an embedding tree for each vertex of , and other PQtrees are connected to in order to encode the dependencies with the cyclic orders of other vertices. Figure 2(b) shows an SPQRdecomposition tree of the graph in Figure 2(a), while Figure 2(c) shows the embedding DAG of the graph in Figure 2(a), which encodes all the embedding constraints for the graph . Note that has only P and Qnodes.
For example, in Figure 2(c) is a PQtree that describes all the possible cyclic orders that the edges incident to can have in a planar embedding of the graph in Figure 2(a): Edges and can be arbitrarily permuted, as well as edges and , while cannot be found between and . The node in and the node in are two consistent Pnodes, which means that in any planar embedding of , if edge is encountered after edge in counterclockwise order around , then edge must be encountered before edge in counterclockwise order around . This constraint depends on the fact that and are the poles of a same triconnected component of , highlighted in gray in Figure 2(a). This constraint is described by the PQtree and by the two edges that are directed from to and from to ; one of these edges is labeled as reversing because the orders of the edges around the two vertices must be opposite to one another. The injective mapping between source PQtrees and sink PQtrees of is not shown in Figure 2(c), but the starting points of the arcs suggest which mappings are suitable. For example, a suitable mapping is between the three leaves , , and of and the three leaves of ; while a suitable mapping between and maps , and to the leaves of .
3 Fixedness and Fixed Constrained Planarity
Bläsius and Rutter in [6] show that normalized instances of Simultaneous FPQOrdering can be solved in quadratic time if they are fixed. In their applications, all instances are already normalized (or have a very simple structure) so that it is easy to verify whether an instance is fixed. The difficulty of applying their result to other contexts is that if the instances are not normalized, it is quite technical to understand the structure of the normalized instance and to check whether it is fixed. In this section we present a new definition of fixedness that does no longer require the normalization as a preliminary step to check whether an instance of Simultaneous FPQOrdering is fixed. This definition, given in Section 3.1 together with the notion of joinable instances, significantly simplifies the application of Simultaneous FPQOrdering. In Section 3.2 we discuss the impact of this definition to efficiently solve a constrained planarity testing problem, called Fixed Constrained Planarity.
3.1 A New Definition of Fixedness
Definition 1
Let be an instance of Simultaneous FPQOrdering and let be a Pnode of an FPQtree that belongs to a node of . The fixedness of is denoted as . Let be the number of children of fixing . If is a source, we define . If is not a source, let be the number of parent nodes of in . For , let be the set of Pnodes of that is fixed by . If for some , then , otherwise . The Pnode is fixed if . Instance is fixed if all Pnodes of FPQtrees are fixed.
We remark that Definition 1 coincides with the notion of fixedness given in [6] (see Section 2) if we restrict ourselves to normalized instances. Namely, in a normalized instance, for , and the maximum vanishes.
Lemma 1 ()
Let be an instance of Simultaneous FPQOrdering and let be the normalization of . Then .
Proof
We recall that normalizing an arc of an instance of Simultaneous FPQOrdering means replacing by , i.e., we first project to the leaves of , which yields a tree whose leaves bijectively correspond to those of and then intersect and to obtain . In this way, any restrictions that imposes on the ordering of the leaves of are transferred to , which thus represents exactly those orders of that can be extended to . The normalization process executes this in topdown fashion for each arc of the instance, thus giving a sequence of instances , where is the number of arcs of [6]. We prove that for , which implies the claim of the lemma.
Assume that is obtained from by normalizing an arc to . Let be a Pnode of . Since is obtained by applying to a sequence of reductions with subsets of leaves whose orders are given by , we have that stems from a single Pnode of . We have the following claim.
Claim
We first show that by using this claim, the inductive step of the lemma follows and then prove the claim. The fixedness of a node depends on the number of children fixing it, as well as, for each parent in which it fixes a Pnode the maximum fixedness of those Pnodes. First observe that, whether a Pnode of some arbitrary FPQtree is fixed by one of its children or not depends solely on the set of leaves and not on any other structural considerations. Since the normalization of an arc does not change the leaf set of any tree, for each Pnode not in the number of children fixing it does not change. For a Pnode of , any child that fixes also fixes the node it stems from, and therefore the number of children fixing is upperbounded by the number of children fixing . Therefore, if a Pnode not in increases its fixedness, this is due to a parent FPQtree containing a Pnode that is fixed by for which increased by the normalization. Traversing the DAG upwards, in this way, we eventually find a Pnode of that is responsible for the increase in fixedness. But then, before the normalization, the fixedness of the node from which stems was used to compute the fixedness of the corresponding child. The claim implies that this fixedness is greater than or equal to the new value that is used to determine the fixedness. This contradicts the assumption that the fixedness of increased.
It remains to prove the claim. As argued above, the numbers and of children of and that fix and , respectively, satisfy . Similarly, let and be the number of parent nodes for which a Pnode is fixed by and , respectively. In particular, and fix Pnodes from the same parent trees . Let and denote the sets of Pnodes of that are fixed by and , respectively. Again, since stems from , it follows that for . Moreover, the fixedness of the nodes in the sets did not increase, since they are not descendants of . Therefore the claim follows.
By Lemma 1, it suffices to check the fixedness of a nonnormalized instance of Simultaneous FPQOrdering to conclude that it can be solved in quadratic time by exploiting [6, Theorems 3.11, 3.16]. We now further simplify the applicability of the result.
Let be an instance of Simultaneous FPQOrdering. We denote by the set of sources of . A solution of an instance of Simultaneous FPQOrdering determines a tuple of cyclic orders . In many cases, we are only interested in the cyclic orders at the sources, and we therefore define has a solution with for . We say that an instance has Pdegree if every node whose FPQtree contains a Pnode has at most parents. Let and be two instances of Simultaneous FPQOrdering such that there exists a bijective mapping between the sources of and the sources of with for each source of . We call and joinable. The join DAG of and is the instance obtained by replacing, for each source node of (and each corresponding source node of ), the nodes (and ) by and identifying the respective nodes of and . By construction, it is .
Lemma 2 ()
Let and be joinable instances of Simultaneous FPQOrdering with Pdegree at most and such that their associated DAGs each have height . If both and are fixed, then is fixed.
Proof
By construction, the height of is , i.e., each node is either a source or a sink. We show that the fixedness of each Pnode of an FPQtree of is at most . We treat the sources and sinks separately. Let be a Pnode of a source of . Since is the intersection of a source of with a source of , it follows that stems from a single Pnode of and from a single Pnode of . Clearly, any child of that fixes must either have fixed or . Hence, since and are fixed.
Let now be a Pnode of a sink of that has at least one parent (otherwise would be a source). Due to the above, Pnodes of all sources are at most fixed. Hence for each Pnode of a parent that is fixed by . Since and have Pdegree at most , has at most two parents. It hence follows that .
3.2 1Fixed Constrained Planarity
Let be a biconnected planar graph, let be a vertex, and let be the edges of incident to . A fixed constraint for is a fixed instance of Simultaneous FPQOrdering such that it has Pdegree at most and it has a single source whose FPQtree has the edges in as its leaves. The following property is implied by [6, Section 4.1].
Property 1
For each vertex of , is a fixed constraint.
Let be an embedding of and let be the cyclic order that induces on the edges around . We say that embedding satisfies constraint if there exists a solution for such that the order of the source is .
Given a graph and a fixed constraint for each vertex of , the Fixed Constrained Planarity testing problem asks whether is fixed constrained planar, i.e., it admits a planar embedding that satisfies all the constraints.
Theorem 3.1 ()
Let be a biconnected planar graph with vertices, and for each let be a fixed constraint. Fixed Constrained Planarity can be tested in time.
Proof
Let be the embedding DAG of , where corresponds bijectively to the rotation systems of the planar embeddings of [6]. The embedding DAG of a vertex is such that corresponds bijectively to the cyclic orders that the planar embeddings of induce around . Let denote the instance of Simultaneous FPQOrdering that is the disjoint union , and observe further that are precisely the rotations at vertices that satisfy all the constraints . Observe further that and are joinable, and are exactly the rotation systems of planar embeddings of that satisfy all the constraints , . By Property 1, both and are fixed, have height and Pdegree at most . Therefore, by Lemma 2 is fixed and by Lemma 1 also the normalization of is fixed. It follows that the normalization of can be solved in time [6, Theorems 3.11, 3.16]. The overall result follows from the fact that and have size linear in and their normalization can be computed in linear time [6, Lemma 3.12].
4 Hybrid Planarity Testing Problems
In this section, we study the interplay between hybrid planarity testing problems and Fixed Constrained Planarity. We consider a variant of the NodeTrix paradigm in Section 4.1, and we study PolyLink representations in Section 4.2, which include special cases of intersectionlink representations [2].
4.1 The RowColumn Independent NodeTrix Planarity Problem
We recall that in a NodeTrix representation each cluster is represented as an adjacency matrix, while the intercluster edges are simple curves connecting the corresponding matrices and not crossing any other matrix [9, 10, 11]. A NodeTrix graph is a flat clustered graph with a NodeTrix representation. For example, Fig. 1(b) is a NodeTrix representation of the graph in Fig. 1(a); note that in this representation for every vertex there are four segments, one for each side of the matrix, to which intercluster edges can be connected. A NodeTrix representation is said to be with fixed sides if the sides of the matrices to which the intercluster edges must be incident are given as part of the input.
The NodeTrix Planarity testing problem with fixed sides is NPhard [9], and it is fixed parameter tractable with respect to the maximum size of clusters and to the branchwidth of the graph obtained by collapsing each cluster into a single vertex, as shown in [10, 12]. NodeTrix Planarity with fixed sides is known to be solvable in linear time when rows and columns are not allowed to be permuted [9]. This naturally raises the question about whether a polynomialtime solution exists also for less constrained versions of NodeTrix Planarity.
We study the scenario in which the permutations of rows and columns can be chosen independently. Namely, we introduce a relaxed version of NodeTrix Planarity with fixed sides, called RowColumn Independent NodeTrix Planarity (RCINT Planarity for short). RCINT Planarity asks whether a flat clustered graph admits a planar NodeTrix representation in the fixed sides scenario, but it allows to permute the rows and the columns independently of one another. A graph for which the RCINT Planarity test is positive is said to be RCINT planar.
The Equipped Frame Graph: We model RCINT Planarity as an instance of Fixed Constrained Planarity defined on a (multi)graph associated with , that we call the equipped frame graph of , denoted as . Graph is obtained from by collapsing each cluster into a single vertex. More precisely, has vertices, each one corresponding to one of the matrices defined by . There is an edge between two vertices and of if and only if there is an edge in between matrices and corresponding to and to , respectively. A NodeTrix graph is biconnected if its equipped frame graph is biconnected and, from now on, we consider biconnected NodeTrix graphs.
Each vertex of is associated with a constraint DAG whose nodes are FPQtrees. More precisely, the source vertex of is an FPQtree consisting of an Fnode with four incident Pnodes; each of such Pnodes describes possible permutations for the rows or for the columns of the matrix . Two Pnodes encode the permutations of the rows (on the left and right handside of ), and the other two Pnodes encode the permutations of the columns (on the top and bottom handside of ). The source of has two adjacent vertices; one of these adjacent vertices is associated with an FPQtree , and the other one is associated with an FPQtree . specifies permutations for the rows of , and specifies permutations for the columns of , that must be respected by the Pnodes of the FPQtree in the root of . We say that and define the coherence between the permutations of the rows and the permutations of the columns, respectively. Fig. 3(a) shows a NodeTrix graph whose clusters have size and Fig. 3(b) shows the constraint DAG associated with vertex of the equipped frame graph of . Note that is RCINT planar but it is not NodeTrix planar with fixed sides: If we require the rows and the columns of to have the same permutation, it is easy to check that either a crossing between and or one between and occurs. Two arcs of Fig. 3(b) are labeled reversing because, for any given permutation of the rows (columns), the rows (columns) are encountered in opposite orders when walking around . Note that is an instance of Simultaneous FPQOrdering.
Property 2
For each vertex of , is a fixed constraint.
Let be the embedding DAG of . Each vertex of is associated with its constraint DAG and its embedding DAG .
Lemma 3 ()
A biconnected NodeTrix graph with fixed sides is RCINT planar if and only if its equipped frame graph is fixed constrained planar.
Proof
Let be a biconnected NodeTrix graph with fixed sides, let be its equipped frame graph, and let be the embedding DAG of . For each , let be a vertex of , let be its constraint DAG, and let be its embedding DAG.
If is fixed constrained planar, admits an embedding that simultaneously satisfies the constraints given by the embedding DAG and the ones given by the constraint DAG , for each . Since each is satisfied, the cyclic orders of the edges around the vertices of describe all the planar combinatorial embeddings of . The constraint DAGs associated with each describe the constraints that allow to replace each vertex of with a matrix whose intercluster edges are incident to the sides as specified by the input. In particular, for each vertex we replace its constraint DAG by a gadget that is built as follows (refer to Fig. 4). The Fnode of the source tree is replaced with a wheel whose external cycle has four vertices , where , one for each edge incident to . Each Pnode that is adjacent to in the source tree is represented in as a vertex that is connected to the corresponding (). Each node is adjacent to Pnodes in . Each (, ) is represented in as a vertex . Finally, for each edge incident to in there is in an edge called spoke incident to . For example, Fig. 4(b) shows the gadget corresponding to the constraint DAG of Fig. 4(a).
Note that the trees and fix the permutations of the children of the two pairs of Pnodes and of , that are guaranteed to be coherent because the constraints described by are satisfied. In , the permutations of the pairs and are fixed consistently. Also, such permutations lead to a planar embedding because they satisfy the embedding DAG .
By performing such a replacement for each vertex and by connecting the spokes of the gadgets that correspond to the same edge, we obtain a planar graph . In order to obtain a planar NodeTrix representation, we replace each gadget by a matrix as follows.
Let , , , , be the vertices that are encountered by walking clockwise along the cycle of the wheel of . Let be the clockwise order of the children of , let be the clockwise order of the children of , let be the clockwise order of the children of , and let be the clockwise order of the children of in . Replace by a matrix whose vertices are placed according to the permutations described for the columns and for the rows. The spokes of that are adjacent to () are connected to on the top side of . Analogously, the spokes of that are adjacent to , , or , are connected to on the right, bottom, or left side of , respectively. By performing this replacement for each gadget of , we obtain a planar NodeTrix representation of the fixed constrained planar graph . It follows that, if is fixed constrained planar, is RCINT planar.
We now show that if is RCINT planar, then is fixed constrained planar. Let be a planar NodeTrix representation of whose rows and columns permutations are independent. Replace each matrix of by a vertex , and connect to it all the intercluster edges that are incident to . We obtain a planar drawing such that the cyclic order of the edges incident to each vertex of reflects the cyclic order of the edges incident to matrix in . Such an order satisfies the constraint DAG of , and it also satisfies the embedding DAG because is planar. Therefore, is fixed constrained planar.
Testing RCINT Planarity: Based on Lemma 3, we shall test whether is RCINT planar by testing whether is fixed constrained planar.
Observe that and have the same leaves, since they describe possible cyclic orders for the same set of intercluster edges, namely those incident to the matrix associated with in , hence and are joinable instances of Simultaneous FPQOrdering. Graph is fixed constrained planar if and only if it admits a planar embedding such that, for each vertex the cyclic order of the edges incident to satisfies both the constraints given by and the ones given by . These constraints are described by the join DAG of and (i.e., ). The following property is implied by Property 1, Property 2, and Lemma 2.
Property 3
For each vertex of , is fixed.
We can now exploit Theorem 3.1, and hence we can decide in time whether is fixed constrained planar, where is the number of vertices of . By Lemma 3, and since constructing may require time, the following theorem holds.
Theorem 4.1 ()
Let be a biconnected NodeTrix graph. RCINT Planarity can be tested in time, where is the number of vertices of .
4.2 PolyLink Planarity Testing
An RCINT planar graph has a planar NodeTrix representation where the intercluster edges are incident to different sides of a gon, and there are constraints that impose the vertices on opposite sides to respect the same permutation. We generalize this type of representation by replacing the gons with gons having an even number of sides.
A PolyLink representation of a flat clustered graph is such that each cluster in is represented as a polygon with an even number of sides. Each side of is associated with its antipodal side along the boundary of . We group the set of vertices of into disjoint subsets; each subset is associated with at least one pair of opposite sides of . Let be a pair of opposite sides of and let be the vertices of associated with . A vertex () is represented by a point on and a point on ; also, when walking clockwise around the vertices along are encountered in opposite order with respect to the vertices along . For each intercluster edge such that is associated with and is associated with , it is specified to which copy of (the one that lies on or the one that lies on ) and to which copy of (the one on or the one on ) edge must be incident.
A PolyLink representation is planar if no two intercluster edges cross. Figure 5(a) shows an example of a planar PolyLink representation.
A flat clustered graph is a PolyLink planar graph if it admits a planar PolyLink representation. We can test a flat clustered graph for PolyLink Planarity by generalizing the approach of Section 4.1. Namely, the constraint DAG associated with a cluster of a PolyLink graph has a source that is an FPQtree, which consists of an Fnode with incident Pnodes, instead of four as in the case of RCINT Planarity. Each of such Pnodes describes the possible permutations of the vertices belonging to a side of the polygon representing . For each pair of sides, the coherence between the order of the vertices belonging to is encoded by means of an FPQtree that is adjacent to the corresponding Pnodes of the source . Figure 5(b) shows the constraint DAG associated with cluster of the graph in Figure 5(a). We say that a flat clustered graph having a PolyLink representation is a biconnected PolyLink graph if its equipped frame graph is biconnected. The same argument used to test RCINT Planarity leads to the following.
Theorem 4.2
Let be a biconnected PolyLink graph. PolyLink Planarity can be tested in time, where is the number of vertices of .
Note that if the sides to which the intercluster edges must be incident are not specified, PolyLink Planarity is NPcomplete. Indeed it becomes equivalent to NodeTrix Planarity with free sides if the polygons have maximum size four and each side is associated with the same set of vertices.
A flat clustered graph that admits an intersectionlink representation is an intersectionlink graph. Let be an intersectionlink graph where is a partition of the vertices of and each cluster of is a clique. We recall that the clique planarity problem asks whether admits an intersectionlink representation where no two intercluster edges cross. In [2] it is proved that if is cliqueplanar, then it admits a canonical intersectionlink representation, i.e., an intersectionlink representation where all vertices in a same cluster are isothetic unit squares whose upperleft corners are aligned along a line of slope one.
Considering a canonical representation of an intersectionlink graph, by walking along the boundary of a cluster of size (with ), the cyclic order of its vertices is such that vertices appear twice and in opposite order, and these two sequences of vertices are separated by two single vertices that appear only once along . We can hence model such an intersectionlink graph as a PolyLink graph where each cluster is a polygon with four sides: A pair of sides contains vertices, while the other two sides contain two vertices, each of which has incident intercluster edges only in one of the two sides.
An instance of clique planarity with fixed sides specifies, for each intercluster edge , the two sides of the unit squares representing and to which is incident. See Figure 6 for an example.
An intersectionlink graph is biconnected if its equipped frame graph is biconnected. By exploiting the relation between PolyLink graphs and intersectionlink graphs, the following corollary holds.
Corollary 1
Let be a biconnected intersectionlink graph. Clique planarity with fixed sides can be tested in time, where is the number of vertices of .
We remark that clique planarity is NPcomplete if the sides to which the intercluster edges are incident are not fixed [2].
5 Open Problems
The research in this paper suggests the following open problems: (i) Our hybrid planarity testing results assume the equipped frame graph to be biconnected. It would be interesting to study the connected case. (ii) What is the complexity of the RowColumn Independent NodeTrix Planarity testing problem in the free sides scenario? (iii) Finally, it would be interesting to validate the RCINT paradigm and the PolyLink paradigm with user studies that compare them with the NodeTrix paradigm. Indeed, the class of RCINT planar graphs is a proper superclass of the planar NodeTrix graphs and if their readability is not significantly worse than in the standard NodeTrix model, the fact that they can be tested in polynomial time could be the starting point for developing new efficient visualization systems.
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