Combinatorics of Triangulations Title: Associate Professor of Information Science

Several combinatorial aspects of triangulations and their generalizations are studied in this thesis. A triangulation of a point con guration and a d-dimensional polyhedron whose vertices are among the points is a decomposition of the polyhedron using d-simplices with vertices among the points. The two main elds triangulations appear are combinatorial geometry in mathematics and computational geometry in information science. The topics connected to triangulations in combinatorial geometry include, polytope theory, Grobner bases of a ne toric ideals, Hilbert bases, generalized hypergeometric functions, and Ehrhart polynomials. Many elds of computational geometry, such as computer graphics, solid modeling, mesh generation, and motion planning, use triangulations extensively. The problems we consider are among the main interests in combinatorial geometry. The topics are on simpli ed basic situations for computational geometry, but are those arising in applications and giving advice for planning algorithms. The objects we study are those around triangulations: (1) triangulations and dissections of 3-polytopes, (2) triangulations (mainly) in the plane, (3) 2-dimensional spheres, and (4) 2-dimensional simplicial complexes. They are from rigid to abstract in this order. The properties of these objects we are interested in are (1) sizes, (2) regularity, which represents the convexity of triangulations, (3) geometric shellings, which are combinatorial topological properties but still expressing convexity, and (4) shellability, extendable shellability, vertex decomposability, which are combinatorial topological properties on incremental construction. They are from basic to ne in this order. The properties become di cult to understand as the dimension becomes higher. We studied the most fundamental cases: basic properties for rigid objects, and ne properties for abstract objects. Four results among our studies are introduced. (See Chapter 1 for the sketch of this thesis.) The rst one is on sizes of decompositions. Dissections are decompositions in which simplices are not required to intersect nicely to form simplicial complexes. They make a superclass of triangulations. We showed size di erence between these two kinds of decompositions for 3-polytopes. We also gave bounds for the size of dissections of 3-polytopes which are similar to the bounds known for triangulations. (Chapter 2) The second result is on nonregular triangulations. We de ned a graph representing the in front/behind order of the maximal dimensional simplices in a triangulation when viewed from a point. The triangulations having some asymmetric cycle in this graph forms a subclass of nonregular triangulations. Thus, we gave a combinatorial subclass of nonregular triangulations, which is of interest, and also studied them further exploiting linear programming duality. (Chapter 3) The third result is on combinatorial topological properties. Here, we studied geometric shellings, shellings becoming line shellings in some combinatorially equivalent polytopes, of 3-polytopes. We showed several su cient conditions for a shelling to be a geometric shelling, which are of interest both in polytope and graph theory. (Chapter 4) The last result also is on combinatorial topological properties. The subject here is the di erence between incremental construction properties, namely, shellability, extendable shellability, and vertex decomposability. By enumerating shellable nonpseudomanifolds of small size, we found smallest examples showing the di erence of these classes, and examples uncovering a relation which was not known. (Chapter 5) An approach in this thesis was to use computers to generate and test examples to study combinatorial geometric aspects of triangulations. We not only devised algorithms of our own, but also made use of established techniques such as integer programming and convex hull computation. (Chapter 6)