Embedding a triangular graph within a given boundary. (English summary) Comput. Aided Geom. Design 28 (2011), no. 6, 349–356. The paper provides a general solution of the following problem: given a 3-vertex-connected planar graph (that is a triangularized area) and an embedding of its boundary vertices in the plane (that is a planar polygon, which is considered as a boundary of a region), can we embed the whole graph into the polygon in a valid way (no overlaps etc.)? This problem was solved for convex polygons in the 1960’s, and for star-shaped polygons in 2006 [S. H. Hong and H. Nagamochi, in Graph-theoretic concepts in computer science, 113–124, Lecture Notes in Comput. Sci., 4271, Springer, Berlin, 2006; MR2290723 (2007i:68096)]. In these cases the solution always exists. In the general case it does not hold, but the paper provides a constructive algorithm, which either shows that no such embedding exists, or provides a valid embedding. The paper is extremely useful in parameterization questions, but has its own beauty as well. Reviewed by Miklós Hoffman
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