Midpoint subdivision generalizes the Lane-Riesenfeld algorithm for uniform tensor product splines and can also be applied to non regular meshes. For example, midpoint subdivision of degree 2 is a specific Doo-Sabin algorithm and midpoint subdivision of degree 3 is a specific Catmull-Clark algorithm. In 2001, Zorin and Schröder were able to prove C1-continuity for midpoint subdivision surfaces a...

Haj\'os conjectured that every graph containing no subdivision of the complete $K_{s+1}$ is properly $s$-colorable. This conjecture was disproved by Catlin. Indeed, maximum chromatic number such graphs $\Omega(s^2/\log s)$. We prove $O(s)$ colors are enough for a weakening this only requires monochromatic component to have bounded size (so-called clustered coloring). Our approach leads more res...

Standard binary subdivision operators may generate surfaces with unbounded curvatures at points corresponding to extraordinary vertices. This defect can be removed by manipulating the eigenvalues of the subdivision operator to impose a bounded curvature spectrum. This procedure may enlarge the support of an extraordinary vertex beyond its two-ring. In the ternary subdivision setting, where mesh...

a {em roman dominating function} on a graph $g = (v ,e)$ is a function $f : vlongrightarrow {0, 1, 2}$ satisfying the condition that every vertex $v$ for which $f (v) = 0$ is adjacent to at least one vertex $u$ for which $f (u) = 2$. the {em weight} of a roman dominating function is the value $w(f)=sum_{vin v}f(v)$. the roman domination number of a graph $g$, denoted by $gamma_r(g)$, equals the...

A non-uniform 3-point ternary interpolatory subdivision scheme with variable subdivision weights is introduced. Its support is computed. The C and C convergence analysis are presented. To elevate its controllability, a modified edition is proposed. For every initial control point on the initial control polygon a shape weight is introduced. These weights can be used to control the shape of the c...

We present an adaptive subdivision scheme for unstructured tetrahedral meshes inspired by the √ 3-subdivision scheme for triangular meshes. Existing tetrahedral subdivision schemes do not support adaptive refinement and have traditionally been driven by the need to generate smooth three-dimensional deformations of solids. These schemes use edge bisections to subdivide tetrahedra, which generate...

We present an adaptive subdivision scheme for unstructured tetrahedral meshes inspired by the √ 3-subdivision scheme for triangular meshes. Existing tetrahedral subdivision schemes do not support adaptive refinement and have traditionally been driven by the need to generate smooth three-dimensional deformations of solids. These schemes use edge bisections to subdivide tetrahedra, which generate...

This paper is concerned with applications of Tutte’s barycentric embedding theorem (1963, Proc. London Math. Soc. 13, 743–768). It presents a method for building isotopies of triangulations in the plane, based on Tutte’s theorem and the computation of equilibrium stresses of graphs by Maxwell–Cremona’s theorem; it also provides a counterexample showing that the analogue of Tutte’s theorem in di...

This paper presents a new unified subdivision scheme that is defined over a k-simplicial complex in n-D space with k ≤ 3. We first present a series of definitions to facilitate topological inquiries during the subdivision process. The scheme is derived from the double (k + 1)-directional box splines over k-simplicial domains. Thus, it guarantees a certain level of smoothness in the limit on a r...