Radial level planarity with fixed embedding

We study level planarity testing of graphs with a fixed combinatorial embedding for three different notions of combinatorial embeddings, namely the level embedding, the upward embedding and the planar embedding. These notions allow for increasing degrees of freedom in their corresponding drawings. For the fixed level embedding there are known and easy to test level planarity criteria. We use these criteria to prove an "untangling" lemma that plays a key role in a simple level planarity test for the case where only the upward embedding is fixed. This test is then adapted to the case where only the planar embedding is fixed. Further, we characterize radial upward planar embeddings, which lets us extend our results to radial level planarity. No algorithms were previously known for these problems

An SPQR-tree-like embedding representation for level planarity

An SPQR-tree is a data structure that efficiently represents all planar embeddings of a biconnected planar graph. It is a key tool in a number of constrained planarity testing algorithms, which seek a planar embedding of a graph subject to some given set of constraints. We develop an SPQR-tree-like data structure that represents all level-planar embeddings of a biconnected level graph with a single source, called the LP-tree, and give a simple algorithm to compute it in linear time. Moreover, we show that LP-trees can be used to adapt three constrained planarity algorithms to the level-planar case by using them as a drop-in replacement for SPQR-trees

SPQR-tree-like embedding representation for level planarity

An SPQR-tree is a data structure that efficiently represents all planar embeddings of a connected planar graph. It is a key tool in a number of constrained planarity testing algorithms, which seek a planar embedding of a graph subject to some given set of constraints. We develop an SPQR-tree-like data structure that represents all level-planar embeddings of a biconnected level graph with a single source, called the LP-tree, and give an algorithm to compute it in linear time. Moreover, we show that LP-trees can be used to adapt three constrained planarity algorithms to the level-planar case by using LP-trees as a drop-in replacement for SPQR-trees

Level-Planar Drawings with Few Slopes

We introduce and study level-planar straight-line drawings with a fixed number of slopes. For proper level graphs (all edges connect vertices of adjacent levels), we give an ( log$^{2}$ / log log )-time algorithm that either finds such a drawing or determines that no such drawing exists. Moreover, we consider the partial drawing extension problem, where we seek to extend an immutable drawing of a subgraph to a drawing of the whole graph, and the simultaneous drawing problem, which asks about the existence of drawings of two graphs whose restrictions to their shared subgraph coincide. We present ($^{4/3}$ log )-time and ($^{10/3}$ log )-time algorithms for these respective problems on proper level-planar graphs. We complement these positive results by showing that testing whether non-proper level graphs admit level-planar drawings with slopes is NP-hard even in restricted cases

Level-Planarity: Transitivity vs. Even Crossings

The strong Hanani-Tutte theorem states that a graph is planar if and only if it can be drawn such that any two edges that do not share an end cross an even number of times. Fulek et al. (2013, 2016, 2017) have presented Hanani-Tutte results for (radial) level-planarity where the $y$-coordinates (distances to the origin) of the vertices are prescribed. We show that the 2-SAT formulation of level-planarity testing due to Randerath et al. (2001) is equivalent to the strong Hanani-Tutte theorem for level-planarity (2013). By elevating this relationship to radial level planarity, we obtain a novel polynomial-time algorithm for testing radial level-planarity in the spirit of Randerath et al

Extending Partial Representations of Circle Graphs in Near-Linear Time

The partial representation extension problem generalizes the recognition problem for geometric intersection graphs. The input consists of a graph G, a subgraph H ⊆ G and a representation H of H. The question is whether G admits a representation G whose restriction to H is H. We study this question for circle graphs, which are intersection graphs of chords of a circle. Their representations are called chord diagrams. We show that for a graph with n vertices and m edges the partial representation extension problem can be solved in O((n+m)α(n+m)) time, where α is the inverse Ackermann function. This improves over an O(n$^{3}$)-time algorithm by Chaplick, Fulek and Klavík [2019]. The main technical contributions are a canonical way of orienting chord diagrams and a novel compact representation of the set of all canonically oriented chord diagrams that represent a given circle graph G, which is of independent interest