Convex drawings of graphs in two and three dimensions (preliminary version)
Proceedings of the twelfth annual symposium on Computational geometry
SIAM Journal on Computing
Graph Drawing: Algorithms for the Visualization of Graphs
Graph Drawing: Algorithms for the Visualization of Graphs
Convex drawings of graphs with non-convex boundary constraints
Discrete Applied Mathematics
Star-Shaped Drawings of Graphs with Fixed Embedding and Concave Corner Constraints
COCOON '08 Proceedings of the 14th annual international conference on Computing and Combinatorics
An algorithm for constructing star-shaped drawings of plane graphs
Computational Geometry: Theory and Applications
Convex drawings of hierarchical planar graphs and clustered planar graphs
Journal of Discrete Algorithms
Convex drawings of 3-connected plane graphs
GD'04 Proceedings of the 12th international conference on Graph Drawing
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A star-shaped drawing of a plane graph is a straight-line drawing such that each inner facial cycle is drawn as a star-shaped polygon, and the outer facial cycle is drawn as a convex polygon. In this paper, given a biconnected plane graph G with fixed plane embedding and a prescribed set of concave corners, we study the following two problems on star-shaped drawings. First, we consider the problem of finding a star-shaped drawing D of G such that only prescribed corners are allowed to become concave corners in D. More specifically, we characterize a necessary and sufficient condition for a subset of prescribed corners to admit such a star-shaped drawing D of G. Our characterization includes Thomassen's characterization of biconnected plane graphs with a prescribed boundary that have convex drawings [24]. We also give a linear-time testing algorithm to test such conditions. Next, given a non-negative cost for each corner in G, we show that a star-shaped drawing D of G with the minimum cost can be found in linear-time, where the cost of a drawing is defined by the sum of costs of concave corners in the drawing.