Subexponential-Time and FPT Algorithms for Embedded Flat Clustered Planarity

Abstract

The C-Planarity problem asks for a drawing of a clustered graph, i.e., a graph whose vertices belong to properly nested clusters, in which each cluster is represented by a simple closed region with no edgeedge crossings, no region-region crossings, and no unnecessary edge-region crossings. We study C-Planarity for embedded flat clustered graphs, graphs with a fixed combinatorial embedding whose clusters partition the vertex set. Our main result is a subexponential-time algorithm to test C-Planarity for these graphs when their face size is bounded. Furthermore, we consider a variation of the notion of embedded tree decomposition in which, for each face, including the outer face, there is a bag that contains every vertex of the face. We show that C-Planarity is fixed-parameter tractable with the embedded-width of the underlying graph and the number of disconnected clusters as parameters.A clustered graph (or c-graph) is a pair C (G, T ) with underlying graph G and inclusion tree T , i.e., a rooted tree whose leaves are the vertices of G. Each internal node µ of T represents a cluster of vertices of G (its leaf descendants) which induces a subgraph G(µ). A c-planar drawing of C (G, T ) ( Fig. 1) consists of a drawing of G and of a representation of each cluster µ as a simple closed region R(µ), i.e., a region homeomorphic to a closed disc, such that: (1) Each region R(µ) contains the drawing of G(µ). (2) For every two clusters µ, ν ∈ T , R(ν) ⊆ R(µ) if and only if ν is a descendant of µ in T . (3) No two edges cross. (4) No edge crosses any region boundary more than once. (5) No two region boundaries intersect.An interesting and challenging line of research in graph drawing concerns the computational complexity of the C-Planarity problem, which asks to test the existence of a c-planar drawing of a c-graph. This problem is notoriously difficult, particularly when (as in Fig. 1) clusters may be disconnected, faces may have unbounded size, and the cluster hierarchy may have multiple nested levels. No known subexponential-time algorithm solves the (general) C-Planarity problem, and it is unknown whether it is NP-complete, although the related problem of splitting as few clusters as possible to make a c-graph c-planar was proved NP-hard [5]. Thus, there is considerable interest in subexponential-time, slice-wise polynomial, and fixed-parameter tractable algorithms, besides polynomial-time algorithms for special cases of C-Planarity.C-Planarity was introduced by Feng, Cohen, and Eades [24], who solved it in quadratic time for the c-connected case when every cluster induces a connected subgraph. Similar results were given by Lengauer [32] using different terminology. Dahlhaus [21] claimed a linear-time algorithm for c-connected C-Planarity (with some details later provided by Cortese et al. [18]). Goodrich et al. [27] gave a cubic-time algorithm for disconnected clusters that satisfy an "extroverted" property, and Gutwenger et al.[28] provided a polynomial-time algorithm for "almost" c-connected inputs. Cornelsen and Wagne...

C-Planarity Testing of Embedded Clustered Graphs with Bounded Dual Carving-Width

“…It is well know that the carving-width cw(δ(H)) of the dual graph δ(H) of a plane graph H with maximum face size (H) and tree-width tw(H) satisfies the relationship cw(δ(H)) ≤ (H)(tw(H) + 2) [12,15]. Therefore, Theorem 1 provides the first 1 polynomialtime algorithm for instances of bounded face size and bounded tree-width, which answers an open question posed by Di Battista and Frati [29,Open Problem (ii)] for instances of bounded tree-width; also, since any n-vertex planar graph has tree-width in O( √ n), it provides an 2 O( √ n) subexponential-time algorithm for instances of bounded face size, which improves the previous 2 O( √ n log n) time bound presented in [26] for such instances. Further implications of Theorem 1 for instances of bounded embedded-width and of bounded dual cut-width are discussed in Section 5.…”

“…In this paper, we consider the parameterized complexity of the C-Planarity Testing problem for embedded c-graphs, i.e., c-graphs with a prescribed combinatorial embedding; see also [13,18,26,46] for previous work in this direction.…”

“…The embedded-width emw(G) of G is the minimum width of any of its tree decompositions that respect the embedding of G. For consistency with other graph-width parameters, in the original definition of this width measure [14] the vertices of the outer face are not required to be in some bag. Here, we adopt the variant presented in [26], where the tree decomposition must also include a bag containing the outer face. We have the following.…”

Planar Drawings with Few Slopes of Halin Graphs and Nested Pseudotrees

“…The family of nested pseudotrees generalizes the well studied 2-outerplanar simply nested graphs and properly includes the Halin graphs [17], the cycle-trees [10], and the cycle-cycles [10]. Simply nested graphs were first introduced by Cimikowski [9], who proved that the innertriangulated ones are Hamiltonian, and have been extensively studied in various contexts, such as universal point sets [1,2], square-contact representations [10], and clustered planarity [11]. Generally, nested pseudotrees have treewidth four and, as such, the best prior upper bound on their planar slope number is the one by Keszegh et al, which is exponential in ∆.…”

Grid Recognition: Classical and Parameterized Computational Perspectives

“…Most (in particular, the early efforts) have been directed at variants of the classic Crossing Minimization problem, introduced by Turán in 1940 [56], parameterized by the number of crossings (see, e.g., [34,46,26,39,40,47]). However, in the past few years, there is an increasing interest in the analysis of a variety of other problems in graph drawing from the perspective of parameterized complexity (see, e.g., [10,2,35,14,38,6,21,20,11,23,49,48] and the upcoming Dagstuhl seminar [1]).…”