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On the existence of a perfect matching for 4-regular graphs derived from quadrilateral meshes. SCI Institute
, 2006
"... In 1891, Peterson [Pet91] proved that every 3-regular bridgeless graph has a perfect matching. It is well-known that the dual of a triangular mesh on a compact manifolds is a 3-regular graph. M. Gopi and D. Eppstein [GE04] use Petersons theorem to solve the problem of constructing strips of triangle ..."
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In 1891, Peterson [Pet91] proved that every 3-regular bridgeless graph has a perfect matching. It is well-known that the dual of a triangular mesh on a compact manifolds is a 3-regular graph. M. Gopi and D. Eppstein [GE04] use Petersons theorem to solve the problem of constructing strips of triangles from triangular meshes on a compact manifold. P. Diaz-Gutierrez and M. Gopi [DG04] elaborate on the creation of strips of quadrilaterals when a perfect matching exists. In this paper, it is shown that the dual of a quadrilateral mesh on a 2-dimensional compact manifold with an even number of quadrilaterals (which is a 4-regular graph) also has a perfect matching. In general, however, not all 4-regular graphs have a perfect matching. Indeed, a counter-example is given that is planar. 1
A Novel Page-Based Data Structure for Interactive Walkthroughs
"... Given a data layout of a large walkthrough scene, we present a novel and simple spatial hierarchy on the disk-pages of the layout that has notable advantages over a conventional spatial hierarchy on the scene geometry. Assume that each disk-page consists of a set of triangles whose bounding boxes ar ..."
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Given a data layout of a large walkthrough scene, we present a novel and simple spatial hierarchy on the disk-pages of the layout that has notable advantages over a conventional spatial hierarchy on the scene geometry. Assume that each disk-page consists of a set of triangles whose bounding boxes are computed. A spatial hierarchy of the walkthrough space is constructed, not with the given scene, but with the bounding boxes of disk-pages. The leaf nodes of the spatial-hierarchy refer directly to the page numbers of the pages of the bounding box it contains. We call this hierarchy on the pages as the disk-page hierarchy. We also propose a self-contained diskpage format that would suit this data structure well. Further, we present a new cache-oblivious graph-based data layout algorithm called the 2-factor layout that would preserve the proximity and orientation properties of the primitives in the layout. Walkthrough experiments have been conducted on a city scene consisting of over 110M triangles. Our system renders this scene on a laptop within a one pixel projection error at over 20 fps with simple texture substitution based simplification of distant objects, and with no explicit data/cache management.
An Algorithm for Computing Simple k-Factors
"... A k-factor of graph G is defined as a k-regular spanning subgraph of G. For instance, a 2-factor of G is a set of cycles that span G. 2-factors have multiple applications in Graph Theory, Computer Graphics, and Computational Geometry [5, 4, 6, 11]. We define a simple 2-factor as a 2-factor without d ..."
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A k-factor of graph G is defined as a k-regular spanning subgraph of G. For instance, a 2-factor of G is a set of cycles that span G. 2-factors have multiple applications in Graph Theory, Computer Graphics, and Computational Geometry [5, 4, 6, 11]. We define a simple 2-factor as a 2-factor without degenerate cycles. In general, simple k-factors are defined as k-regular spanning subgraphs where no edge is used more than once. We propose a new algorithm for computing simple k-factors for all values of k ≥ 2. 1
A Novel Page-Based Data Structure for Interactive Walkthroughs
"... Given a data layout of a large walkthrough scene, we present a novel and simple spatial hierarchy on the disk-pages of the layout that has notable advantages over a conventional spatial hierarchy on the scene geometry. Assume that each disk-page consists of a set of triangles whose bounding boxes ar ..."
Abstract
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Given a data layout of a large walkthrough scene, we present a novel and simple spatial hierarchy on the disk-pages of the layout that has notable advantages over a conventional spatial hierarchy on the scene geometry. Assume that each disk-page consists of a set of triangles whose bounding boxes are computed. A spatial hierarchy of the walkthrough space is constructed, not with the given scene, but with the bounding boxes of disk-pages. The leaf nodes of the spatial-hierarchy refer directly to the page numbers of the pages of the bounding box it contains. We call this hierarchy on the pages as the disk-page hierarchy. We also propose a self-contained diskpage format that would suit this data structure well. Further, we present a new cache-oblivious graph-based data layout algorithm called the 2-factor layout that would preserve the proximity and orientation properties of the primitives in the layout. Walkthrough experiments have been conducted on a city scene consisting of over 110M triangles. Our system renders this scene on a laptop within a one pixel projection error at over 20 fps with simple texture substitution based simplification of distant objects, and with no explicit data/cache management.
An Algorithm for Computing Simple k-Factors
"... A k-factor of graph G is defined as a k-regular spanning subgraph of G. For instance, a 2-factor of G is a set of cycles that span G. 2-factors have multiple applications in Graph Theory, Computer Graphics, and Computational Geometry [5, 4, 6, 14]. We define a simple 2-factor as a 2-factor without d ..."
Abstract
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A k-factor of graph G is defined as a k-regular spanning subgraph of G. For instance, a 2-factor of G is a set of cycles that span G. 2-factors have multiple applications in Graph Theory, Computer Graphics, and Computational Geometry [5, 4, 6, 14]. We define a simple 2-factor as a 2-factor without degenerate cycles. In general, simple k-factors are defined as k-regular spanning subgraphs where no edge is used more than once. We propose a new algorithm for computing simple k-factors for all values of k ≥ 2. Key words: graph algorithms, graph factors, k-factors, simple k-factors, 2-factors 1.
