Irreducible triangulations of surfaces with boundary

A triangulation of a surface is irreducible if no edge can be contracted to produce a triangulation of the same surface. In this paper, we investigate irreducible triangulations of surfaces with boundary. We prove that the number of vertices of an irreducible triangulation of a (possibly non-orientable) surface of genus g>=0 with b>=0 boundaries is O(g+b). So far, the result was known only for surfaces without boundary (b=0). While our technique yields a worse constant in the O(.) notation, the present proof is elementary, and simpler than the previous ones in the case of surfaces without boundary

Triangulating the Real Projective Plane

We consider the problem of computing a triangulation of the real projective plane P2, given a finite point set S={p1, p2,..., pn} as input. We prove that a triangulation of P2 always exists if at least six points in S are in general position, i.e., no three of them are collinear. We also design an algorithm for triangulating P2 if this necessary condition holds. As far as we know, this is the first computational result on the real projective plane

Testing Convexity and Acyclicity, and New Constructions for Dense Graph Embeddings

Property testing, especially that of geometric and graph properties, is an ongoing area of research. In this thesis, we present a result from each of the two areas. For the problem of convexity testing in high dimensions, we give nearly matching upper and lower bounds for the sample complexity of algorithms have one-sided and two-sided error, where algorithms only have access to labeled samples independently drawn from the standard multivariate Gaussian. In the realm of graph property testing, we give an improved lower bound for testing acyclicity in directed graphs of bounded degree. Central to the area of topological graph theory is the genus parameter, but the complexity of determining the genus of a graph is poorly understood when graphs become nearly complete. We summarize recent progress in understanding the space of minimum genus embeddings of such dense graphs. In particular, we classify all possible face distributions realizable by minimum genus embeddings of complete graphs, present new constructions for genus embeddings of the complete graphs, and find unified constructions for minimum triangulations of surfaces