We prove that every internally 4-connected non-planar bipartite graph has an odd K3,3 subdivision; that is, a subgraph obtained from K3,3 by replacing its edges by internally disjoint odd paths with the same ends. The proof gives rise to a polynomial-time algorithm to find such a subdivision. (A bipartite graph G is internally 4-connected if it is 3-connected, has at least five vertices, and th...

We describe the structure of 2-connected non-planar toroidal graphs with no K3,3-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 K5-subdivision in [A. Gagarin and W. Kocay, “Embedding graphs containing K5-subdivisions”...

§

‎let $g=(v(g),e(g))$ be a graph‎, ‎$gamma_t(g)$. let $ooir(g)$ be the total domination and oo-irredundance number of $g$‎, ‎respectively‎. ‎a total dominating set $s$ of $g$ is called a $textit{total perfect code}$ if every vertex in $v(g)$ is adjacent to exactly one vertex of $s$‎. ‎in this paper‎, ‎we show that if $g$ has a total perfect code‎, ‎then $gamma_t(g)=ooir(g)$‎. ‎as a consequence, ...

In this, an algorithm to produce ternary m-point (for any integer m ≥ 2) approximating subdivision scheme has been introduced that can generate the families of C limiting curves. The proposed scheme has developed using the uniform B-spline blending functions and its convergence is analyzed by Laurent polynomial method. It is concluded that the existing 2-point, 3-point and 5-point ternary appro...

Subdivision rules create sequences of nested cell structures on CWcomplexes, and they frequently arise from groups. In this paper, we develop several tools for classifying subdivision rules. We give a criterion for a subdivision rule to represent a Gromov hyperbolic space, and show that a subdivision rule for a hyperbolic group determines the Gromov boundary. We give a criterion for a subdivisi...

In geometric models with high-valence vertices, current subdivision wavelets may not deal with the special cases well for good visual effect of multiresolution surfaces. In this paper, we present the novel biorthogonal polar subdivision wavelets, which can efficiently perform wavelet analysis to the control nets with polar structures. The polar subdivision can generate more natural subdivision ...

An eternal $m$-secure set of a graph $G = (V,E)$ is aset $S_0subseteq V$ that can defend against any sequence ofsingle-vertex attacks by means of multiple-guard shifts along theedges of $G$. A suitable placement of the guards is called aneternal $m$-secure set. The eternal $m$-security number$sigma_m(G)$ is the minimum cardinality among all eternal$m$-secure sets in $G$. An edge $uvin E(G)$ is ...

The precision set, approximation order and Hölder exponent are derived for each of the univariate subdivision schemes described in [3] and [4].

Both the 4-point and the uniform cubic B-spline subdivisions double the number of vertices of a closed-loop polygon P and produce sequences of vertices fj and bj respectively. We study the J-spline subdivision scheme Js, introduced by Maillot and Stam, which blends these two methods to produce vertices of the form vj=(1–s)fj+sbj. Iterative applications of Js yield a family of limit curves, the ...