Hierarchical graphs and clustered graphs are useful non-classical graph models for structured relational information. Hierarchical graphs are graphs with layering structures; clustered graphs are graphs with recursive clustering structures. Both have applications in CASE tools, software visualization, and VLSI design. Drawing algorithms for hierarchical graphs have been well investigated. However, the problem of straight-line representation has not been solved completely. In this paper, we answer the question: does every planar hierarchical graph admit a planar straight-line hierarchical drawing? We present an algorithm that constructs such drawings in linear time. Also, we answer a basic question for clustered graphs, that is, does every planar clustered graph admit a planar straight-line drawing with clusters drawn as convex polygons? We provide a method for such drawings based on our algorithm for hierarchical graphs.

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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.

In this paper we develop a characterisation of minimal non-planarity of level graphs. We show that a level minimal non-planar (LMNP) graph is completely characterised by either a tree, a level non-planar cycle or a level planar cycle with certain path augmentations. We discuss the usefulness of these characterisations in the context of an branch-and-cut Integer Linear Programming implementation of the Maximum Level Planar Subgraph (MLPS) problem and conjecture that the inequalities associated with level minimal non-planar subgraphs are facet-defining for the MLPS polytope. 1 Introduction Graph layout by Integer Linear Programming (ILP) has gained remarkable success recently. The method has, generally, the following framework: ffl compute the planar subgraph having the maximum number of edges by ILP; ffl compute the layout of the planar subgraph by an exact polynomial-time algorithm; and, ffl add non-planar edges to the layout. This approach has various advantages compared to other ...

A wide number of practical applications would benefit from automatically generated graphical representations of database schemas, in which tables are represented by boxes, and table attributes correspond to distinct stripes inside each table. Links, connecting attributes of two different tables, represent referential constraints or join relationships, and may attach arbitrarily to the left or to the right side of the stripes representing the attributes. To our knowledge no drawing technique is available to automatically produce diagrams in such strongly constrained drawing convention. In this paper we provide a polynomial time algorithm for solving this problem and test its efficiency and effectiveness against a large test suite. Also, we describe an implementation of a system that uses such an algorithm and we study the main methodological problems we faced in developing such a technology.

A graph is planar if it can be drawn on the plane with vertices at unique locations and no edge intersections except at the vertex endpoints. Due to the wealth of interest from the computer science community, there are a number of remarkable but complex O(n) planar embedding algorithms. This paper presents an O(n) planar embedding algorithm that avoids a number of the complexities of prior approaches (an early version of this work was presented at the January 1999 Symposium on Discrete Algorithms).

A graph is planar if it can be drawn on the plane with vertices at unique locations and no edge intersections except at the vertex endpoints. Recent research eorts have produced new algorithms for solving planarity-related problems. Shih and Hsu proposed a linear-time algorithm based on a data structure they named PC-tree, which is similar to but much simpler than a PQ-tree. However, their presentation does not explain in detail how to implement the routines that manipulate a PC-tree, and there are some nontrivial correctness and run-time issues that were not addressed in their paper. So it is far from trivial to derive a proper linear-time implementation from their description. This paper presents additions to the theoretical framework of the PC-tree algorithm that are necessary to achieve correctness and linear running time. A linear-time implementation that addresses the issues raised in this paper was developed in the LEDA platform and is available.

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