On the Parameterized Complexity of Layered Graph Drawing

2001

We study the problem of producing hierarchical drawings of layered graphs when some pairs of edges are not allowed to cross. We show that deciding on the existence of a drawing satisfying at least k constraints from a given set of non-crossing constraints is NP-complete even if the graph is 2-layered and even when the permutation of the vertices on one side of the bipartition is fixed.

Journal of Discrete Algorithms, 2006

We study the problem of computing hierarchical drawings of layered graphs when some pairs of edges are not allowed to cross. We show that deciding the existence of a drawing satisfying at least k non-crossing constraints from a given set is NP-hard, even if the graph is 2-layered and even when the permutation of the vertices on one side of the bipartition is fixed. We then propose simple constant-ratio approximation algorithms for the optimization version of the problem, which requires to find a maximum realizable subset of constraints, and we discuss how to extend the well-known hierarchical approach for creating layered drawings of directed graphs so as to minimize the number of edge crossings while maximizing the number of satisfied constraints.

Journal of Graph Algorithms and Applications, 2005

Sugiyama's algorithm for layered graph drawing is very popular and commonly used in practical software. The extensive use of dummy vertices to break long edges between non-adjacent layers often leads to unsatisfying performance. The worst-case running-time of Sugiyama's approach is O(|V ||E| log |E|) requiring O(|V ||E|) memory, which makes it unusable for the visualization of large graphs. By a conceptually simple new technique we are able to keep the number of dummy vertices and edges linear in the size of the graph without increasing the number of crossings. We reduce the worst-case time complexity of Sugiyama's approach by an order of magnitude to O((|V | + |E|) log |E|) requiring O(|V | + |E|) space.

Computational Geometry, 2015

We initiate the study of the following problem: Given a non-planar graph G and a planar subgraph S of G, does there exist a straight-line drawing Γ of G in the plane such that the edges of S are not crossed in Γ by any edge of G? We give positive and negative results for different kinds of connected spanning subgraphs S of G. Moreover, in order to enlarge the subset of instances that admit a solution, we consider the possibility of bending the edges of G not in S; in this setting we discuss different trade-offs between the number of bends and the required drawing area.

Lecture Notes in Computer Science, 2018

In this paper we present a new approach to visualize directed graphs and their hierarchies that completely departs from the classical four-phase framework of Sugiyama and computes readable hierarchical visualizations that contain the complete reachability information of a graph. Additionally, our approach has the advantage that only the necessary edges are drawn in the drawing, thus reducing the visual complexity of the resulting drawing. Furthermore, most problems involved in our framework require only polynomial time. Our framework offers a suite of solutions depending upon the requirements, and it consists of only two steps: (a) the cycle removal step (if the graph contains cycles) and (b) the channel decomposition and hierarchical drawing step. Our framework does not introduce any dummy vertices and it keeps the vertices of a channel vertically aligned. The time complexity of the main drawing algorithms of our framework is O(kn), where k is the number of channels, typically much smaller than n (the number of vertices). G. Ortali-This author's research was performed in part while he was visiting the University of Crete.

Journal of Graph Algorithms and Applications, 2013

A straight-line drawing of a planar graph G is a planar drawing of G, where each vertex is mapped to a point on the Euclidean plane and each edge is drawn as a straight line segment. A segment in a straight-line drawing is a maximal set of edges that form a straight line segment. A minimum-segment drawing of G is a straightline drawing of G, where the number of segments is the minimum among all possible straight-line drawings of G. In this paper we prove that it is NP-complete to determine whether a plane graph G has a straight-line drawing with at most k segments, where k ≥ 3. We also prove that the problem of deciding whether a given partial drawing of G can be extended to a straight-line drawing with at most k segments is NP-complete, even when G is an outerplanar graph. Finally, we investigate a worst-case lower bound on the number of segments required by straight-line drawings of arbitrary spanning trees of a given planar graph.

2010

The most popular method of drawing directed graphs is to place vertices on a set of horizontal or concentric levels, known as level drawings. Level drawings are well studied in Graph Drawing due to their strong application for the visualization of hierarchy in graphs. There are two drawing conventions: horizontal drawings use a set of parallel lines and radial drawings use a set of concentric circles.

ACM Journal of Experimental Algorithmics, 2005

We propose two fast heuristics for solving the NP-hard problem of graph layering with the minimum width and consideration of dummy nodes. Our heuristics can be used at the layer-assignment phase of the Sugiyama method for drawing of directed graphs. We evaluate our heuristics by comparing them to the widely used fast-layering algorithms in an extensive computational study with nearly 6000 input graphs. We also demonstrate how the well-known longest-path and Coffman–Graham algorithms can be used for finding narrow layerings with acceptable aesthetic properties.

Lecture Notes in Computer Science, 2010

A rectilinear drawing is an orthogonal grid drawing without bends, possibly with edge crossings, without any overlapping between edges, between vertices, or between edges and vertices. Rectilinear drawings without edge crossings (planar rectilinear drawings) have been extensively investigated in graph drawing. Testing rectilinear planarity of a graph is NP-complete . Restricted cases of the planar rectilinear drawing problem, sometimes called the "no-bend orthogonal drawing problem", have been well studied (see, for example, ).

Lecture Notes in Computer Science, 2011

We study the complexity of the problem of finding nonplanar rectilinear drawings of graphs. This problem is known to be NPcomplete. We consider natural restrictions of this problem where constraints are placed on the possible orientations of edges. In particular, we show that if each edge has prescribed direction "left", "right", "down" or "up", the problem of finding a rectilinear drawing is polynomial, while finding such a drawing with the minimum area is NP-complete. When assigned directions are "horizontal" or "vertical" or a cyclic order of the edges at each vertex is specified, the problem is NP-complete. We show that these two NP-complete cases are fixed parameter tractable in the number of vertices of degree 3 or 4.