Subdivision surface method is a series of iterative operation adopts a certain subdivision formula for an initial grid, obtains the smooth limits surface finally, and can dispose any arbitrary complex topology grid. At present most of the subdivision algorithm are 1-4 subdivisions and as the number of subdivision increase, the grid grow so toorapid in the number of patch that it is difficult fo...

Let m≥1 be an integer and G a graph with m edges. We say that has antimagic orientation if D bijection τ:A(D)→{1,…,m} such no two vertices in have the same vertex-sum under τ, where of vertex v τ is sum labels all arcs entering minus leaving v. Hefetz et al. (2010) conjectured every connected admits orientation. The conjecture was confirmed for certain classes graphs as regular graphs, minimum ...

There has been an emerging interest in developing an approximation theory for manifold-valued functions. In this paper, we address the following fundamental problem: Let M be a manifold with a metric d. For each smoothness factor r > 0 and approximation order R > 0, is there an approximation operator Ah = Ah;r,R that maps samples of any f : R→ M on a grid of size h to an approximant fh = Ah(f |...

In the Eternal Domination game, a team of guard tokens initially occupies a dominating set on a graph G. A rioter then picks a node without a guard on it and attacks it. The guards defend against the attack: one of them has to move to the attacked node, while each remaining one can choose to move to one of his neighboring nodes. The new guards’ placement must again be dominating. This attack-de...

It has been shown that one triangulation of a set of points can be converted to any other triangulation of the same set of points by a sequence of edge flip operations. In this paper we consider a tesselation of a set of points consisting of convex cells, a convex subdivision, and explore the notion of flipping edges from one convex subdivision of the points to another, where both subdivisions ...

We propose an adaptive geometry compression method based on 4-point interpolatory subdivision schemes. It can work on digital curves of arbitrary dimensions. With the geometry compression method, a digital curve is adaptively compressed into several segments with different compression levels. Each segment is a 4-point subdivision curve with a subdivision step. In the meantime, we provide high-s...

A set D of vertices of a graph G is a dominating set if every vertex in V \D is adjacent to some vertex in D. The domination number γ(G) of G is the minimum cardinality of a dominating set of G. The domination subdivision number of G is the minimum number of edges that must be subdivided (where each edge in G can be subdivided at most once) in order to increase the domination number. Arumugam h...

Given a graph G = (V, E), the subdivision of an edge e = uv ∈ E(G) means the substitution of the edge e by a vertex x and the new edges ux and xv. The domination subdivision number of a graph G is the minimum number of edges of G which must be subdivided (where each edge can be subdivided at most once) in order to increase the domination number. Also, the domination multisubdivision number of G...

Committee in Charge 1996 i Acknowledgements I a m d eeply indebted to m y advisors, Karl Levitt a n d Manfred Ruschitzka. Without their help and guidance, I was not able to nish this work. Special thanks to Karl for his precious time a n d e n ergy, a n d h i s p a tience in teaching m e not only how t o perform research, but also how to write proposals, deal with sponsors, give presentations, ...

A tropical complete intersection curve C ⊆ Rn+1 is a transversal intersection of n smooth tropical hypersurfaces. We give a formula for the number of vertices of C given by the degrees of the tropical hypersurfaces. We also compute the genus of C (defined as the number of independent cycles of C) when C is smooth and connected. 1 Notation and definitions We work over the tropical semifield Rtr=...