This paper investigates the following question: Given an integer grid phi, where phi is a proper subset of the integer plane or a proper subset of the integer 3d space, which graphs admit straight-line crossingfree drawings with vertices located at the grid points of phi? We characterize the trees that can be drawn on a two dimensional c * n × k grid, where k and c are given integer constants, and on a two dimensional grid consisting of k parallel horizontal lines of infinite length. Motivated by the results on the plane we investigate restrictions of the integer grid in 3 dimensions and show that every outerplanar graph with n vertices can be drawn crossing-free with straight lines in linear volume on a grid called a prism. This prism consists of 3n integer grid points and is universal -- it supports all outerplanar graphs of n vertices. This is the first algorithm that computes crossing-free straight line 3d drawings in linear volume for a non-trivial family of planar graphs. We also show that there exist planar graphs that cannot be drawn on the prism and that extension to a n × 2 × 2 integer grid, called a box, does not admit the entire class of planar graphs.

We prove that every n-vertex graph G with pathwidth pw(G) has a three-dimensional straight-line grid drawing with O(pw(G) n) volume. Thus for

In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A k-stack (resp...

A queue layout of a graph consists of a total order of the vertices, and a partition of the edges into queues, such that no two edges in the same queue are nested. The minimum number of queues in a queue layout of a graph is its queue-number. A three-dimensional (straight- line grid) drawing of a graph represents the vertices by points in Z and the edges by non-crossing line-segments. This paper contributes three main results: (1) It is proved that the minimum volume of a certain type of three-dimensional drawing of a graph G is closely related to the queue-number of G. In particular, if G is an n-vertex member of a proper minor-closed family of graphs (such as a planar graph), then G has a O(1) O(1) O(n) drawing if and only if G has O(1) queue-number.

A graph with a given partition of the vertices on k concentric circles is radial level planar if there is a vertex permutation such that the edges can be routed strictly outwards without crossings. Radial level planarity extends level planarity, where the vertices are placed on k horizontal lines and the edges are routed strictly downwards without crossings. The extension is characterised by rings, which are level non-planar biconnected components. Our main results are linear time algorithms for radial level planarity testing and for computing an embedding. We introduce PQR-trees as a new data structure where R-nodes and associated templates for their manipulation are introduced to deal with rings. Our algorithms extend level planarity testing and embedding algorithms which use PQ-trees.

Consider a graph G drawn in the plane so that each vertex lies on a distinct horizontal line ℓj = {(x, j) | x ∈ R}. The bijection φ that maps the set of n vertices V to a set of distinct horizontal lines ℓj forms a labeling of the vertices. Such a graph G with the labeling φ is called an n-level graph and is said to be n-level planar if it can be drawn with straight-line edges and no crossings while keeping each vertex on its own level. In this paper, we consider the class of trees that are n-level planar regardless of their labeling. We call such trees unlabeled level planar (ULP). Our contributions are three-fold. First, we provide a complete characterization of ULP trees in terms of a pair of forbidden subtrees. Second, we show how to draw ULP trees in linear time. Third, we provide a linear time recognition algorithm for ULP trees.

A bipartite graph is biplanar if the vertices can be placed on two parallel lines (layers) in the plane such that there are no edge crossings when edges are drawn straight. The 2-Layer Planarization problem asks if k edges can be deleted from a given graph G so that the remaining graph is biplanar. This problem is NP-complete, as is the 1-Layer Planarization problem in which the permutation of the vertices in one layer is fixed. We give the following fixed parameter tractability results: an O(k ·6 k +|G|) algorithm for 2-Layer Planarization and an O(3 k ·|G|) algorithm for 1-Layer Planarization, thus achieving linear time for fixed k.

We give an O(ϕ k · n 2) fixed parameter tractable algorithm for the 1-SIDED CROSSING MINIMIZATION problem. The constant ϕ in the running time is the golden ratio ϕ = (1 + √ 5)/2 ≈ 1.618. The constant k is the parameter of the problem: the number of allowed edge crossings.

We exhibit a small problem kernel for the one-sided crossing minimization problem. This problem plays an important role in graph drawing algorithms based on the Sugiyama layering approach. Moreover, we improve on the search tree algorithm developed in [7] and derive an O(1.4656 k + kn 2) algorithm for this problem, where k upperbounds the number of tolerated edge crossings in the drawings of an n-vertex graph.

This paper surveys parameterized complexity results for hard geometric algorithmic problems. It includes fixed-parameter tractable problems in graph drawing, geometric graphs, geometric covering and several other areas, together with an overview of the algorithmic techniques used. Fixed-parameter intractability results are surveyed as well. Finally, we give some directions for future research.