A New Perspective on Clustered Planarity as a Combinatorial Embedding Problem

Abstract

The clustered planarity problem (c-planarity) asks whether a hierarchically clustered graph admits a planar drawing such that the clusters can be nicely represented by regions. We introduce the cd-tree data structure and give a new characterization of c-planarity. It leads to efficient algorithms for c-planarity testing in the following cases. (i) Every cluster and every co-cluster (complement of a cluster) has at most two connected components. (ii) Every cluster has at most five outgoing edges.Moreover, the cd-tree reveals interesting connections between c-planarity and planarity with constraints on the order of edges around vertices. On one hand, this gives rise to a bunch of new open problems related to c-planarity, on the other hand it provides a new perspective on previous results.

C-Planarity of Embedded Cyclic c-Graphs

“…The result also follows from a work by Gutwenger et al [17]. Beyond two clusters a polynomial time algorithm for c-planarity was obtained only in special cases, e.g., [8,16,17,20,21], and most recently in [6,7]. Cortese et al [9] shows that c-planarity is solvable in…”

Bounded Embeddings of Graphs in the Plane

“…For example, the ordering corresponding to the graph in Figure 6 is (4, 6), (3,6), (2,6), (4,7), (3,7), (2,7). E (s, b).…”

“…Thus, our goal is to reduce the problem to the problem of finding an ordering of a finite set that satisfies constraints given by a collection of PC-trees. 6 The construction of the corresponding instance I of the simultaneous PC-ordering for the given (G, γ) is inspired by [5,Section 4.2]. Thus, the instance consists of a star T C having a P-node in the center (consistency tree), and a collection of embedding trees T 0 , .…”

“…A subdivided star (on the left) with the center v, and some restrictions on the set of rotation at v (on the right) corresponding to the intervals (s α , b α ), (s α,1 , b α ) and (s, b). We have {e 0 , e 5 , e6 } = E(s α,1 , b α ) ⊆ E(s, b) = {e 0 , e 2 ,e 5 , e 6 } and {e 3 , e 4 } = E (s, b) ⊆ E (s α,1 , b α ) = {e 3 , e 4 , e 1 , e 2 }. Thus, by removing e 0 from E(s, b) we obtain the same restrictions on the rotation at v.Lemma 5.5.…”

Clustered Planarity with Pipes

“…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.…”