Constrained Point-Set Embeddability of Planar Graphs
Di Giacomo, Emilio and Didimo, Walter and Liotta, Giuseppe and Meijer, Henk and Wismath, Stephen (2009) Constrained Point-Set Embeddability of Planar Graphs. In: Graph Drawing 16th International Symposium, GD 2008, September 21- 24, 2008, Heraklion, Crete, Greece , pp. 360-371 (Official URL: http://dx.doi.org/10.1007/978-3-642-00219-9_35).
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This paper starts the investigation of a constrained version of the pointset embeddability problem. Let G = (V , E ) be a planar graph with n vertices, G′ = (V ′ , E ′ ) a subgraph of G, and S a set of n distinct points in the plane. We study the problem of computing a point-set embedding of G on S subject to the constraint that G′ is drawn with straight-line edges. Different drawing algorithms are presented that guarantee small curve complexity of the resulting drawing, i.e. a small number of bends per edge. It is proved that: (i) If G′ is an outerplanar graph and S is any set of points in convex position, a point-set embedding of G on S can be computed such that the edges of E \ E ′ have at most 4 bends each. (ii) If S is any set of points in general position and G′ is a face of G or if it is a simple path, the curve complexity of the edges of E \ E ′ is at most 8. (iii) If S is in general position and G′ is a set of k disjoint paths, the curve complexity of the edges of E \ E ′ is O(2k ).
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