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
Convex drawings of hierarchical planar graphs and clustered planar graphs
Journal of Discrete Algorithms
Convex drawings of graphs with non-convex boundary
WG'06 Proceedings of the 32nd international conference on Graph-Theoretic Concepts in Computer Science
Convex drawings of 3-connected plane graphs
GD'04 Proceedings of the 12th international conference on Graph Drawing
Theoretical Computer Science
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A straight-line planar drawing of a plane graph is called a convex drawing if every facial cycle is drawn as a convex polygon. Convex drawings of graphs is a well-established aesthetic in graph drawing, however not all planar graphs admit a convex drawing. Tutte [W.T. Tutte, Convex representations of graphs, Proc. of London Math. Soc. 10 (3) (1960) 304-320] showed that every triconnected plane graph admits a convex drawing for any given boundary drawn as a convex polygon. Thomassen [C. Thomassen, Plane representations of graphs, in: Progress in Graph Theory, Academic Press, 1984, pp. 43-69] gave a necessary and sufficient condition for a biconnected plane graph with a prescribed convex boundary to have a convex drawing. In this paper, we initiate a new notion of star-shaped drawing of a plane graph as a straight-line planar 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. A star-shaped drawing is a natural extension of a convex drawing, and a new aesthetic criteria for drawing planar graphs in a convex way as much as possible. We give a sufficient condition for a given set A of corners of a plane graph to admit a star-shaped drawing whose concave corners are given by the corners in A, and present a linear time algorithm for constructing such a star-shaped drawing.