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Area Minimization for Grid Visibility Representation of Hierarchically Planar Graphs
 Proceedings of the 5th International Conference on Computing and Combinatorics (COCOON ’99
, 1999
"... . Hierarchical graphs are an important class of graphs for modelling many real applications in software and information visualization. In this paper, we shall investigate the computational complexity of constructing minimum area grid visibility representations of hierarchically planar graphs. Fi ..."
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. Hierarchical graphs are an important class of graphs for modelling many real applications in software and information visualization. In this paper, we shall investigate the computational complexity of constructing minimum area grid visibility representations of hierarchically planar graphs. Firstly, we provide a quadratic algorithm that minimizes the drawing area with respect to a fixed planar embedding. This implies that the area minimization problem is polynomial time solvable restricted to the class of graphs whose planar embeddings are unique. Secondly, we show that the area minimization problem is generally NPhard. Keywords: Graph Drawing, Hierarchically Planar Graph, Visibility Representation, Drawing Area. 1 Introduction Automatic graph drawing plays an important role in many modern computerbased applications, such as CASE tools, software and information visualization, VLSI design, visual data mining, and internet navigation. Directed acyclic graphs are an importa...
Study of Proper Hierarchical Graphs on a Grid
"... Abstract—Hierarchical planar graph embedding (sometimes called level planar graphs) is widely recognized as a very important task in diverse fields of research and development. Given a proper hierarchical planar graph, we want to find a geometric position of every vertex (layout) in a straightline ..."
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Abstract—Hierarchical planar graph embedding (sometimes called level planar graphs) is widely recognized as a very important task in diverse fields of research and development. Given a proper hierarchical planar graph, we want to find a geometric position of every vertex (layout) in a straightline grid drawing without any edgeintersection. An additional objective is to minimize the area of the rectangular grid in which G is drawn with more aesthetic embedding. In this paper we propose several ideas to find an embedding of G in a rectangular grid with area, ( �1) × (k1), where � is the number of vertices in the longest level and k is the number of levels in G.) Usually, we use one of some aesthetic criteria (such as drawing area minimization, minimizing the number of edge crossings, symmetry, bends minimization or distributing the vertices uniformly) in order to make the layout of a graph readable and understandable [6],[7]. Reducing the number of edge crossings or distributing the vertices uniformly have been proposed and evaluating goodness of drawing based on these criteria has been reported [8], [9], [10]. Many works have been published area requirements for drawing hierarchically planar graphs [6],[7], [8], [11]. Keywordslevel graphs; hierarchical graphs; algorithms; graph drawing. I.
HananiTutte and Monotone Drawings
"... Abstract. A drawing of a graph is xmonotone if every edge intersects every vertical line at most once and every vertical line contains at most one vertex. Pach and Tóth showed that if a graph has an xmonotone drawing in which every pair of edges crosses an even number of times, then the graph has ..."
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Abstract. A drawing of a graph is xmonotone if every edge intersects every vertical line at most once and every vertical line contains at most one vertex. Pach and Tóth showed that if a graph has an xmonotone drawing in which every pair of edges crosses an even number of times, then the graph has an xmonotone embedding in which the xcoordinates of all vertices are unchanged. We give a new proof of this result and strengthen it by showing that the conclusion remains true even if adjacent edges are allowed to cross oddly. This answers a question posed by Pach and Tóth. Moreover, we show that an extension of this result for graphs with nonadjacent pairs of edges crossing oddly fails even if there exists only one such pair in a graph. 1