Hierarchy of surface models and irreducible triangulations

AbstractGiven a triangulated closed surface, the problem of constructing a hierarchy of surface models of decreasing level of detail has attracted much attention in computer graphics. A hierarchy provides view-dependent refinement and facilitates the computation of parameterization. For a triangulated closed surface of n vertices and genus g, we prove that there is a constant c>0 such that if n>c·g, a greedy strategy can identify Θ(n) topology-preserving edge contractions that do not interfere with each other. Further, each of them affects only a constant number of triangles. Repeatedly identifying and contracting such edges produces a topology-preserving hierarchy of O(n+g2) size and O(logn+g) depth. Although several implementations exist for constructing hierarchies, our work is the first to show that a greedy algorithm can efficiently compute a hierarchy of provably small size and low depth. When no contractible edge exists, the triangulation is irreducible. Nakamoto and Ota showed that any irreducible triangulation of an orientable 2-manifold has at most max{342g−72,4} vertices. Using our proof techniques we obtain a new bound of max{240g,4}

Irreducible Triangulations are Small

A triangulation of a surface is \emph{irreducible} if there is no edge whose contraction produces another triangulation of the surface. We prove that every irreducible triangulation of a surface with Euler genus $g\geq1$ has at most $13g-4$ vertices. The best previous bound was $171g-72$.Comment: v2: Referees' comments incorporate

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

Subgraph densities in a surface

Given a fixed graph $H$ that embeds in a surface $\Sigma$, what is the maximum number of copies of $H$ in an $n$-vertex graph $G$ that embeds in $\Sigma$? We show that the answer is $\Theta(n^{f(H)})$, where $f(H)$ is a graph invariant called the `flap-number' of $H$, which is independent of $\Sigma$. This simultaneously answers two open problems posed by Eppstein (1993). When $H$ is a complete graph we give more precise answers.Comment: v4: referee's comments implemented. v3: proof of the main theorem fully rewritten, fixes a serious error in the previous version found by Kevin Hendre

Hierarchy of Surface Models and Irreducible Triangulations

Sheung-Hung Poon Given a triangulated closed surface, the problem of constructing a hierarchy of surface models of decreasing level of detail has attracted much attention in computer graphics. A hierarchy provides viewdependent refinement and facilitates the computation of parameterization. For a triangulated closed surface of ¢ vertices and genus £ , we prove that there is a constant ¤¦¥¨ § such that if ¢©¥¨¤��� £ , a greedy strategy can identify ����¢� � topology-preserving edge contractions that do not interfere with each other. Further, each of them affects only a constant number of triangles. Repeatedly identifying and contracting such edges produces a topology-preserving hierarchy of ����¢���£��� � size and ����������¢���£�� depth. In practice, the genus £ is very small when compared with ¢ for large models and the selection of edges can be enhanced by measuring the error of their contractions using some known heuristics. Although several implementations exist for constructing hierarchies, our work is the first to show that a greedy algorithm can efficiently compute a hierarchy of provably small size and low depth. When no contractible edge exists, the triangulation is irreducible. Nakamoto and Ota showed that any irreducible triangulation of an orientable 2-manifold has at most ��������������£����������� � vertices. Using our proof techniques we obtain a new bound of ������������§�£����� �