Manhattan-Geodesic Embedding of Planar Graphs
Katz, Bastian and Krug, Marcus and Rutter, Ignaz and Wolﬀ, Alexander (2010) Manhattan-Geodesic Embedding of Planar Graphs. In: Graph Drawing 17th International Symposium, GD 2009, September 22-25, 2009, Chicago, IL, USA , pp. 207-218 (Official URL: http://dx.doi.org/10.1007/978-3-642-11805-0_21).
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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 N P-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 N P-hard even for perfect matchings, but without the grid restriction, we solve the matching problem eﬃciently.
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