Study of Proper Hierarchical Graphs on a Grid

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 straight-line grid drawing without any edge-intersection. 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) × (k-1), where  is the number of vertices in the longest level and k is the number of levels in G.) Keywords-level graphs; hierarchical graphs; algorithms; graph drawing.