Clustered Planarity with Pipes

Abstract

We study the version of the C-PLANARITY problem in which edges connecting the same pair of clusters must be grouped into pipes, which generalizes the STRIP PLANARITY problem. We give algorithms to decide several families of instances for the two variants in which the order of the pipes around each cluster is given as part of the input or can be chosen by the algorithm. IntroductionVisualizing clustered graphs is a challenging task with several applications in the analysis of networks that exhibit a hierarchical structure. The most established criterion for a readable visualization of these graphs has been formalized in the notion of c-planarity, introduced by Feng, Cohen, and Eades [13] in 1995. Given a clustered graph C (G, T ) (c-graph), that is, a graph G equipped with a recursive clustering T of its vertices, the C-PLANARITY problem asks whether there exist a planar drawing of G and a representation of each cluster as a topological disk enclosing all and only its vertices, such that no "unnecessary" crossings occur between disks and edges, or between disks. Ever since its introduction, this problem has been attracting a great deal of research. However, the question regarding its computational complexity withstood the attack of several powerful algorithmic tools, such as the Hanani-Tutte theorem [14,16], the SPQR-tree machinery [10], and the Simultaneous PQ-ordering framework [6].The clustering of a c-graph C (G, T ) is described by a rooted tree T whose leaves are the vertices of G and whose each internal node µ different from the root represents a cluster containing all and only the leaves of the subtree of T rooted at µ. A c-graph is flat if T has height 2. The clusters-adjacency graph G A of a flat c-graph is the graph obtained from the c-graph by contracting each cluster into a single vertex, and by removing multi-edges and loops.Cortese et al.[11] introduced a variant of C-PLANARITY for flat c-graphs, which we call C-PLANARITY WITH EMBEDDED PIPES. The input of this problem is a flat c-graph C (G, T ) together with a planar drawing of its clusters-adjacency graph G A , in which vertices of G A are represented by disks and edges of G A by pipes connecting the disks. The goal is then to produce a c-planar drawing of C (G, T ) in which each vertex of G lies inside the disk representing the cluster it belongs to and each inter-cluster edge of G is drawn inside the corresponding pipe. In [11] this problem is solved when the underlying graph G is a cycle. Chang, Erickson, and Xu [9] observed that in this case the problem is equivalent to determining whether a closed walk of length n in a simple plane graph is weakly simple, and improved the time complexity to O(n log n). The special case of the problem in which the clusters-adjacency graph is a path while G can be any planar graph, which is known by the name of STRIP PLANARITY, has also been studied. Polynomial-time algorithms for this problem have been presented when the underlying graph has a fixed planar embedding [2] and when it is a tree [14].We remark th...

Subexponential-Time and FPT Algorithms for Embedded Flat Clustered Planarity

“…FPT algorithms have also been investigated [10,15]. For additional special cases, see, e.g., [2,3,4,7,14,23].A c-graph is flat when no non-trivial cluster is a subset of another, so T has only three levels: the root, the clusters, and the leaves. Flat C-Planarity can be solved in polynomial time for embedded c-graphs with at most 5 vertices per face [22,26] or at most two vertices of each cluster per face [13], for embedded c-graphs in which each cluster induces a subgraph with at most two connected components [30], and for c-graphs with two clusters [9,26,29] or three clusters [1].…”

“…FPT algorithms have also been investigated [10,15]. For additional special cases, see, e.g., [2,3,4,7,14,23].…”

Recognizing Weak Embeddings of Graphs

“…Weak embeddings of graphs also generalize various graph visualization models such as the recently introduced strip planarity [3] and level planarity [18]; and can be seen as a special case [2] of the notoriously difficult clustered-planarity (for short, c-planarity) [8,12,13], whose tractability remains elusive despite many attempts by leading researchers.…”

“…Let φ (1) : G (1) → H (1) be the input instance of pipeExpansion(uv), φ (2) : G (1) → H (2) be the instance obtained by contracting uv and φ (3) : G (3) → H (3) be the instance after clusterExpansion( uv ). By the previous argument, the total orders of pipe-edges π (2) ( uv w ) of all pipes uv w can be obtained from a combinatorial representation π (3) in O (deg( uv )) time. These orders also correspond to π (1) (uw ) and π (1) (vx ) for pipes uw and vx in φ (1) where w v and x u.…”

Atomic Embeddability, Clustered Planarity, and Thickenability

“…The problem can be seen as a hierarchical variant of planarity testing; a problem for which a linear time algorithm has been known for a long time [30]. In the extensive literature devoted to c-planarity and its variants, the complexity status of only restricted special cases has been established, most notably in [2,5,17,27], see also the somewhat outdated survey [16]. The c-planarity problem is formally stated as follows.…”