The Cost of Perfection for Matchings in Graphs

Perfect matchings and maximum weight matchings are two fundamental combinatorial structures. We consider the ratio between the maximum weight of a perfect matching and the maximum weight of a general matching. Motivated by the computer graphics application in triangle meshes, where we seek to convert a triangulation into a quadrangulation by merging pairs of adjacent triangles, we focus mainly on bridgeless cubic graphs. First, we characterize graphs that attain the extreme ratios. Second, we present a lower bound for all bridgeless cubic graphs. Third, we present upper bounds for subclasses of bridgeless cubic graphs, most of which are shown to be tight. Additionally, we present tight bounds for the class of regular bipartite graphs

A simple and flexible framework to adapt dynamic meshes

Many graphics applications represent deformable surfaces through dynamic meshes. To be consistent during deformations, the dynamic meshes require an adaptation process. In this paper we present a simple and flexible framework to adapt dynamic meshes following deformable surfaces. Our scheme combines normal and tangential geometric corrections with refinement and simplification resolution control. It works with different surface descriptions, and supports application-specific criteria. We also introduce a stochastic sampling approach to measure the geometric error approximation. As an example, we couple our framework with numerical simulations, such as a particle level-set method. (C) 2008 Elsevier Ltd. All rights reserved.32214114