Manhattan-geodesic embedding of planar graphs

Abstract

In this paper, we explore a new convention for drawing graphs, the (Manhattan-) geodesic drawing convention. It requires that edges are drawn as interior-disjoint monotone chains of axis-parallel line segments, that is, as geodesics with respect to the Manhattan metric. First, we show that geodesic embeddability on the grid is equivalent to 1-bend embeddability on the grid. For the latter question an efficient algorithm has been proposed. Second, we consider geodesic point-set embeddability where the task is to decide whether a given graph can be embedded on a given point set. We show that this problem is -hard. In contrast, we efficiently solve geodesic polygonization-the special case where the graph is a cycle. Third, we consider geodesic point-set embeddability where the vertex-point correspondence is given. We show that on the grid, this problem is -hard even for perfect matchings, but without the grid restriction, we solve the matching problem efficiently. © 2010 Springer-Verlag.