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

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 $$O(r(n^2) + n^2)$$ O ( r ( n 2 ) + n 2 ) for general, possibly non-flat, instances.

Bisected vertex leveling of plane graphs: Braid index, arc index and delta diagrams

In this paper, we introduce a bisected vertex leveling of a plane graph. Using this planar embedding, we present elementary proofs of the well-known upper bounds in terms of the minimal crossing number on braid index [Formula: see text] and arc index [Formula: see text] for any knot or non-split link [Formula: see text], which are [Formula: see text] and [Formula: see text]. We also find a quadratic upper bound of the minimal crossing number of delta diagrams of [Formula: see text].

A Large Set of Torus Obstructions and How They Were Discovered

We outline the progress made so far on the search for the complete set of torus obstructions and also consider practical algorithms for torus embedding and their implementations. We present the set of obstructions that are known to-date and give a brief history of how these graphs were found. We also describe a nice algorithm for embedding graphs on the torus which we used to verify previous results and add to the set of torus obstructions. Although it is still exponential in the order of the graph, the algorithm presented here is relatively simple to describe and implement and fast-in-practice for small graphs.It parallels the popular quadratic planar embedding algorithm of Demoucron, Malgrange, and Pertuiset.