Two-Layer Planarization Parameterized by Feedback Edge Set

Given an undirected graph G and an integer k ≥ 0, the NP-hard 2-Layer Planarization problem asks whether G can be transformed into a forest of caterpillar trees by removing at most k edges. Since transforming G into a forest of caterpillar trees requires breaking every cycle, the size f of a minimum feedback edge set is a natural parameter with f ≤ k. We improve on previous fixed-parameter tractability results with respect to k by presenting a problem kernel with O(f) vertices and edges and a new search-tree based algorithm, both with about the same worst-case bounds for f as the previous results for k, although we expect f to be smaller than k for a wide range of input instances

Automated lattice drawing

Abstract. Lattice diagrams, known as Hasse diagrams, have played an ever increasing role in lattice theory and fields that use lattices as a tool. Initially regarded with suspicion, they now play an important role in both pure lattice theory and in data representation. Now that lattices can be created by software, it is important to have software that can automatically draw them. This paper covers: – The role and history of the diagram. – What constitutes a good diagram. – Algorithms to produce good diagrams. Recent work on software incorporating these algorithms into a drawing program will also be covered. An ordered set P = (P, ≤) consists of a set P and a partial order relation ≤ on P. That is, the relation ≤ is reflexive (x ≤ x), transitive (x ≤ y and y ≤ z imply x ≤ z) and antisymmetric (x ≤ y and y ≤ x imply x = y). If P is finite there is a unique smallest relation ≺, known as the cover or neighbor relation, whose transitive, reflexive closure is ≤. (Graph theorists call this the transitive reduct of ≤.) A Hasse diagram of P is a diagram of the acyclic graph (P, ≺) where the edges are straight line segments and, if a < b in P, then the vertical coordinate for a is less than the one for b. Because of this second condition arrows are omitted from the edges in the diagram. A lattice is an ordered set in which every pair of elements a and b has a least upper bound, a ∨ b, and a greatest lower bound, a ∧ b, and so also has a Hasse diagram. These Hasse diagrams 1 are an important tool for researchers in lattice theory and ordered set theory and are now used to visualize data. This paper deals the special issues involved in such diagrams. It gives several approaches that have been used to automatically draw such diagrams concentrating on a three dimension force algorithm especially adapted for ordered sets that does particularly well. We begin with some examples. 1 In the second edition of his famous book on lattice theory [3] Birkhoff says these diagrams are called Hasse diagrams because of Hasse’s effective use of them but that they go back at least to H. Vogt, Résolution algébrique des équation, Paris, 1895.

On the parameterized complexity of layered graph drawing

We consider graph drawings in which vertices are assigned to layers and edges are drawn as straight line-segments between vertices on adjacent layers. We prove that graphs admitting crossing-free h-layer drawings (for fixed h) have bounded pathwidth. We then use a path decomposition as the basis for a linear-time algorithm to decide if a graph has a crossing-free h-layer drawing (for fixed h). This algorithm is extended to solve related problems, including allowing at most k crossings, or removing at most r edges to leave a crossing-free drawing (for fixed k or r). If the number of crossings or deleted edges is a non-fixed parameter then these problems We consider graph drawings in which vertices are assigned to layers and edges are drawn as straight line-segments between vertices on adjacent layers. We prove that graphs admitting crossing-free h-layer drawings (for fixed h) have bounded pathwidth. We then use a path decomposition as the basis for a linear-time algorithm to decide if a graph has a crossing-free h-layer drawing (for fixed h). This algorithm is extended to solve related problems, including allowing at most k crossings, or removing at most r edges to leave a crossing-free drawing (for fixed k or r). If the number of crossings or deleted edges is a non-fixed parameter then these problems are NP-complete. For each setting, we can also permit downward drawings of directed graphs and drawings in which edges may span multiple layers, in which case either the total span or the maximum span of edges can be minimized. In contrast to the Sugiyama method for layered graph drawing, our algorithms do not assume a preassignment of the vertices to layers