DeepAI

# Simultaneous FPQ-Ordering and Hybrid Planarity Testing

We study the interplay between embedding constrained planarity and hybrid planarity testing. We consider a constrained planarity testing problem, called 1-Fixed Constrained Planarity, and prove that this problem can be solved in quadratic time for biconnected graphs. Our solution is based on a new definition of fixedness that makes it possible to simplify and extend known techniques about Simultaneous PQ-Ordering. We apply these results to different variants of hybrid planarity testing, including a relaxation of NodeTrix Planarity with fixed sides, that allows rows and columns to be independently permuted.

• 32 publications
• 31 publications
• 18 publications
07/11/2019

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## 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 inter-cluster edge while an edge with both end-vertices in a same cluster is an intra-cluster edge. A hybrid representation of is a drawing of the graph that adopts different visualization paradigms to represent the clusters and to represent the inter-cluster 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 inter-cluster edges are drawn according to the node-link paradigm. NodeTrix representations have been introduced to visually explore non-planar 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).

Intersection-link 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 intra-cluster edge connecting them [2, 3]. Fig. 1(d) is an intersection-link 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 inter-cluster 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 inter-cluster 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 FPQ-trees (a variant of PQ-trees). 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 FPQ-Ordering. -Fixed Constrained Planarity generalizes the partially PQ-constrained 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 multi-graph obtained by collapsing the clusters to single vertices is biconnected. It may be worth recalling that NodeTrix Planarity with fixed sides is NP-complete in general, but it is linear-time 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 intersection-link planarity, called clique planarity with fixed sides, can be solved in quadratic time. Note that clique planarity is known to be NP-complete 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 Row-Column 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

PQ-trees: A PQ-tree is a data structure that represents a family of permutations on a set of elements [7]. In a PQ-tree, each element is represented by a leaf node, and each non-leaf node is either a P-node or a Q-node. The children of a P-node can be arbitrarily permuted, while the order of the children of a Q-node is fixed up to a reversal. Three main operations are defined on PQ-trees [6, 7]. Let be a PQ-tree and let be the set of its leaves. Given , the projection of to , denoted as , is a PQ-tree 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 degree-2 nodes together with both incident edges are iteratively replaced by single edges. The reduction of with , denoted as , is a PQ-tree that represents only the orders represented by where the leaves of are consecutive. A Q-node in can determine the orientation of several Q-nodes of , while if we consider a P-node in , there is exactly one P-node in that depends on . We say that stems from . Given two PQ-trees and , the intersection of and , denoted as , is a PQ-tree 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 PQ-Ordering: An instance of Simultaneous PQ-Ordering [6] is a DAG of PQ-trees 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 PQ-trees 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 PQ-Ordering 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 PQ-Ordering is normalized if, for each arc and for each internal node , tree contains exactly one node that is fixed by . Every instance of Simultaneous PQ-Ordering 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 P-node of a PQ-tree 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 P-node is -fixed if . Also, instance is -fixed if all the P-nodes of any PQ-tree are -fixed.

FPQ-trees: An FPQ-tree is a PQ-tree where, for some of the Q-nodes, the reversal of the permutation described by their children is not allowed. To distinguish these Q-nodes from the regular Q-nodes, we call them F-nodes [12]. The study of Bläsius and Rutter on Simultaneous PQ-Ordering also considers the case in which the permutations described by some of the Q-nodes are totally fixed, hence the results given in [6] for Simultaneous PQ-Ordering also hold when the nodes of the input DAG are FPQ-trees. In the rest of the paper we talk about Simultaneous FPQ-Ordering to emphasize the presence of F-nodes, 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 SPQR-decomposition 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 SPQR-decomposition tree of . We can “translate” an SPQR-decomposition tree of into a set of PQ-trees (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 PQ-trees by describing and exploiting the relation between PQ-trees and SPQR-trees [6, Section ]. The obtained DAG of PQ-trees is the embedding DAG , whose size is linear in the size of the SPQR-tree , and thus linear in the size of itself. has an embedding tree for each vertex of , and other PQ-trees are connected to in order to encode the dependencies with the cyclic orders of other vertices. Figure 2(b) shows an SPQR-decomposition 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 Q-nodes.

For example, in Figure 2(c) is a PQ-tree 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 P-nodes, which means that in any planar embedding of , if edge is encountered after edge in counter-clockwise order around , then edge must be encountered before edge in counter-clockwise 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 PQ-tree 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 PQ-trees and sink PQ-trees 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 1-Fixed Constrained Planarity

Bläsius and Rutter in [6] show that normalized instances of Simultaneous FPQ-Ordering 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 FPQ-Ordering is -fixed. This definition, given in Section 3.1 together with the notion of joinable instances, significantly simplifies the application of Simultaneous FPQ-Ordering. 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 FPQ-Ordering and let be a P-node of an FPQ-tree 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 P-nodes of that is fixed by . If for some , then , otherwise . The P-node is -fixed if . Instance is -fixed if all P-nodes of FPQ-trees 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 FPQ-Ordering and let  be the normalization of . Then .

###### Proof

We recall that normalizing an arc of an instance of Simultaneous FPQ-Ordering 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 top-down 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 P-node 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 P-node  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 P-node the maximum fixedness of those P-nodes. First observe that, whether a P-node  of some arbitrary FPQ-tree 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 P-node not in the number of children fixing it does not change. For a P-node  of , any child that fixes  also fixes the node  it stems from, and therefore the number of children fixing  is upper-bounded by the number of children fixing . Therefore, if a P-node  not in increases its fixedness, this is due to a parent FPQ-tree containing a P-node  that is fixed by  for which increased by the normalization. Traversing the DAG upwards, in this way, we eventually find a P-node  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 P-node is fixed by  and , respectively. In particular, and  fix P-nodes from the same parent trees . Let  and  denote the sets of P-nodes 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 non-normalized instance of Simultaneous FPQ-Ordering 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 FPQ-Ordering. We denote by  the set of sources of . A solution of an instance  of Simultaneous FPQ-Ordering 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 P-degree if every node whose FPQ-tree contains a P-node has at most parents. Let  and  be two instances of Simultaneous FPQ-Ordering 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 FPQ-Ordering with P-degree 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 P-node of an FPQ-tree of is at most . We treat the sources and sinks separately. Let  be a P-node of a source of . Since  is the intersection of a source of with a source of , it follows that  stems from a single P-node  of and from a single P-node  of . Clearly, any child of that fixes  must either have fixed  or . Hence, since  and  are -fixed.

Let now be a P-node of a sink of that has at least one parent (otherwise would be a source). Due to the above, P-nodes of all sources are at most -fixed. Hence  for each P-node  of a parent that is fixed by . Since and  have P-degree at most , has at most two parents. It hence follows that .

### 3.2 1-Fixed 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 FPQ-Ordering such that it has P-degree at most  and it has a single source whose FPQ-tree 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 FPQ-Ordering 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 P-degree 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 intersection-link representations [2].

### 4.1 The Row-Column Independent NodeTrix Planarity Problem

We recall that in a NodeTrix representation each cluster is represented as an adjacency matrix, while the inter-cluster 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 inter-cluster edges can be connected. A NodeTrix representation is said to be with fixed sides if the sides of the matrices to which the inter-cluster edges must be incident are given as part of the input.

The NodeTrix Planarity testing problem with fixed sides is NP-hard [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 polynomial-time 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 Row-Column Independent NodeTrix Planarity (RCI-NT Planarity for short). RCI-NT 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 RCI-NT Planarity test is positive is said to be RCI-NT planar.

The Equipped Frame Graph: We model RCI-NT 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 FPQ-trees. More precisely, the source vertex of is an FPQ-tree consisting of an F-node with four incident P-nodes; each of such P-nodes describes possible permutations for the rows or for the columns of the matrix . Two P-nodes encode the permutations of the rows (on the left and right hand-side of ), and the other two P-nodes encode the permutations of the columns (on the top and bottom hand-side of ). The source of has two adjacent vertices; one of these adjacent vertices is associated with an FPQ-tree , and the other one is associated with an FPQ-tree . specifies permutations for the rows of , and specifies permutations for the columns of , that must be respected by the P-nodes of the FPQ-tree 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 RCI-NT 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 FPQ-Ordering.

###### 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 RCI-NT 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 inter-cluster 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 F-node of the source tree is replaced with a wheel whose external cycle has four vertices , where , one for each edge incident to . Each P-node 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 P-nodes 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 P-nodes 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 RCI-NT planar.

We now show that if is RCI-NT 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 inter-cluster 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 RCI-NT Planarity: Based on Lemma 3, we shall test whether is RCI-NT 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 inter-cluster edges, namely those incident to the matrix associated with in , hence and are joinable instances of Simultaneous FPQ-Ordering. 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. RCI-NT Planarity can be tested in -time, where is the number of vertices of .

An RCI-NT planar graph has a planar NodeTrix representation where the inter-cluster 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 inter-cluster 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 inter-cluster 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 FPQ-tree, which consists of an F-node with incident P-nodes, instead of four as in the case of RCI-NT Planarity. Each of such P-nodes 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 FPQ-tree that is adjacent to the corresponding P-nodes 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 RCI-NT 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 inter-cluster edges must be incident are not specified, PolyLink Planarity is NP-complete. 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 intersection-link representation is an intersection-link graph. Let be an intersection-link 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 intersection-link representation where no two inter-cluster edges cross. In [2] it is proved that if is clique-planar, then it admits a canonical intersection-link representation, i.e., an intersection-link representation where all vertices in a same cluster are isothetic unit squares whose upper-left corners are aligned along a line of slope one.

Considering a canonical representation of an intersection-link 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 intersection-link 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 inter-cluster edges only in one of the two sides.

An instance of clique planarity with fixed sides specifies, for each inter-cluster edge , the two sides of the unit squares representing and to which is incident. See Figure 6 for an example.

An intersection-link graph is biconnected if its equipped frame graph is biconnected. By exploiting the relation between PolyLink graphs and intersection-link graphs, the following corollary holds.

###### Corollary 1

Let be a biconnected intersection-link graph. Clique planarity with fixed sides can be tested in time, where is the number of vertices of .

We remark that clique planarity is NP-complete if the sides to which the inter-cluster 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 Row-Column Independent NodeTrix Planarity testing problem in the free sides scenario? (iii) Finally, it would be interesting to validate the RCI-NT paradigm and the PolyLink paradigm with user studies that compare them with the NodeTrix paradigm. Indeed, the class of RCI-NT 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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