Abstract
For a clustered graph, i.e, a graph whose vertex set is recursively partitioned into clusters, the C-Planarity Testing problem asks whether it is possible to find a planar embedding of the graph and a representation of each cluster as a region homeomorphic to a closed disk such that (1) the subgraph induced by each cluster is drawn in the interior of the corresponding disk, (2) each edge intersects any disk at most once, and (3) the nesting between clusters is reflected by the representation, i.e., child clusters are properly contained in their parent cluster. The computational complexity of this problem, whose study has been central to the theory of graph visualization since its introduction in 1995 [Feng, Cohen, and Eades, Planarity for clustered graphs, ESA’95], has only been recently settled [Fulek and Tóth, Atomic Embeddability, Clustered Planarity, and Thickenability, to appear at SODA’20]. Before such a breakthrough, the complexity question was still unsolved even when the graph has a prescribed planar embedding, i.e, for embedded clustered graphs. We show that the C-Planarity Testing problem admits a single-exponential single-parameter FPT (resp., XP) algorithm for embedded flat (resp., non-flat) clustered graphs, when parameterized by the carving-width of the dual graph of the input. These are the first FPT and XP algorithms for this long-standing open problem with respect to a single notable graph-width parameter. Moreover, the polynomial dependency of our FPT algorithm is smaller than the one of the algorithm by Fulek and Tóth. In particular, our algorithm runs in quadratic time for flat instances of bounded treewidth and bounded face size. To further strengthen the relevance of this result, we show that an algorithm with running time O(r(n)) for flat instances whose underlying graph has pathwidth 1 would result in an algorithm with running time O(r(n)) for flat instances and with running time O(r(n^2) + n^2) for general, possibly non-flat, instances.
Highlights
Many real-world data exhibit an intrinsic hierarchical structure that can be captured in the form of clustered graphs, i.e., graphs equipped with a recursive clustering of their vertices
A c-planar drawing of a c-graph C(G, T) is a planar drawing of G together with a representation of each cluster in T as a region D( ) homeomorphic to a closed disc such that: (1) for each cluster in T, region D( ) contains the drawing of the subgraph G[V ] of G induced by V ; (2) for every two clusters and in T, it holds D(η) ⊆ D(μ) if and only if is a descendant of in T ; (3) each edge crosses the boundary of any cluster disk at most once; and (4) the boundaries of no two cluster disks intersect
Since h ≤ n, Theorem 1.2 immediately translates into a slice-wise polynomial (XP) algorithm, parameterized by the dual carving width, for C-Planarity Testing of embedded non-flat c-graphs
Summary
Many real-world data exhibit an intrinsic hierarchical structure that can be captured in the form of clustered graphs, i.e., graphs equipped with a recursive clustering of their vertices. 2, we show that the C-Planarity Testing problem retains its complexity when restricted to instances of bounded path-width and to connected instances of bounded tree-width Such a result, which holds even for general, non-embedded, c-graphs, implies that the goal of devising an algorithm parameterized by graphwidth parameters that are within a constant factor from tree-width (e.g., branchwidth [66]) or that are bounded by path-width (e.g., tree-width, rank-width [64], boolean-width [1], and clique-width [32]) and with a dependency on the input size which improves upon the one in [47] appears to be a significant algorithmic challenge. Since h ≤ n , Theorem 1.2 immediately translates into a slice-wise polynomial (XP) algorithm, parameterized by the dual carving width, for C-Planarity Testing of embedded non-flat c-graphs
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