Single-Strip Triangulation of Manifolds with Arbitrary Topology

Triangle strips have been widely used for efficient rendering. It is NP-complete to test whether a given triangulated model can be represented as a single triangle strip, so many heuristics have been proposed to partition models into few long strips. In this paper, we present a new algorithm for creating a single triangle loop or strip from a triangulated model. Our method applies a dual graph matching algorithm to partition the mesh into cycles, and then merges pairs of cycles by splitting adjacent triangles when necessary. New vertices are introduced at midpoints of edges and the new triangles thus formed are coplanar with their parent triangles, hence the visual fidelity of the geometry is not changed. We prove that the increase in the number of triangles due to this splitting is 50% in the worst case, however for all models we tested the increase was less than 2%. We also prove tight bounds on the number of triangles needed for a single-strip representation of a model with holes on its boundary. Our strips can be used not only for efficient rendering, but also for other applications including the generation of space filling curves on a manifold of any arbitrary topology.Comment: 12 pages, 10 figures. To appear at Eurographics 200

Characterization and enumeration of toroidal K_{3,3}-subdivision-free graphs

We describe the structure of 2-connected non-planar toroidal graphs with no K_{3,3}-subdivisions, using an appropriate substitution of planar networks into the edges of certain graphs called toroidal cores. The structural result is based on a refinement of the algorithmic results for graphs containing a fixed K_5-subdivision in [A. Gagarin and W. Kocay, "Embedding graphs containing K_5-subdivisions'', Ars Combin. 64 (2002), 33-49]. It allows to recognize these graphs in linear-time and makes possible to enumerate labelled 2-connected toroidal graphs containing no K_{3,3}-subdivisions and having minimum vertex degree two or three by using an approach similar to [A. Gagarin, G. Labelle, and P. Leroux, "Counting labelled projective-planar graphs without a K_{3,3}-subdivision", submitted, arXiv:math.CO/0406140, (2004)].Comment: 18 pages, 7 figures and 4 table

Partial-Matching and Hausdorff RMS Distance Under Translation: Combinatorics and Algorithms

We consider the RMS distance (sum of squared distances between pairs of points) under translation between two point sets in the plane, in two different setups. In the partial-matching setup, each point in the smaller set is matched to a distinct point in the bigger set. Although the problem is not known to be polynomial, we establish several structural properties of the underlying subdivision of the plane and derive improved bounds on its complexity. These results lead to the best known algorithm for finding a translation for which the partial-matching RMS distance between the point sets is minimized. In addition, we show how to compute a local minimum of the partial-matching RMS distance under translation, in polynomial time. In the Hausdorff setup, each point is paired to its nearest neighbor in the other set. We develop algorithms for finding a local minimum of the Hausdorff RMS distance in nearly linear time on the line, and in nearly quadratic time in the plane. These improve substantially the worst-case behavior of the popular ICP heuristics for solving this problem.Comment: 31 pages, 6 figure

Edge Roman domination on graphs

An edge Roman dominating function of a graph $G$ is a function $f\colon E(G) \rightarrow \{0,1,2\}$ satisfying the condition that every edge $e$ with $f(e)=0$ is adjacent to some edge $e'$ with $f(e')=2$. The edge Roman domination number of $G$, denoted by $\gamma'_R(G)$, is the minimum weight $w(f) = \sum_{e\in E(G)} f(e)$ of an edge Roman dominating function $f$ of $G$. This paper disproves a conjecture of Akbari, Ehsani, Ghajar, Jalaly Khalilabadi and Sadeghian Sadeghabad stating that if $G$ is a graph of maximum degree $\Delta$ on $n$ vertices, then $\gamma_R'(G) \le \lceil \frac{\Delta}{\Delta+1} n \rceil$. While the counterexamples having the edge Roman domination numbers $\frac{2\Delta-2}{2\Delta-1} n$, we prove that $\frac{2\Delta-2}{2\Delta-1} n + \frac{2}{2\Delta-1}$ is an upper bound for connected graphs. Furthermore, we provide an upper bound for the edge Roman domination number of $k$-degenerate graphs, which generalizes results of Akbari, Ehsani, Ghajar, Jalaly Khalilabadi and Sadeghian Sadeghabad. We also prove a sharp upper bound for subcubic graphs. In addition, we prove that the edge Roman domination numbers of planar graphs on $n$ vertices is at most $\frac{6}{7}n$, which confirms a conjecture of Akbari and Qajar. We also show an upper bound for graphs of girth at least five that is 2-cell embeddable in surfaces of small genus. Finally, we prove an upper bound for graphs that do not contain $K_{2,3}$ as a subdivision, which generalizes a result of Akbari and Qajar on outerplanar graphs