Let G be a graph and let S(G), M(G), T(G) the subdivision, middle, total of G, respectively. dim(G), edim(G), mdim(G) metric dimension, edge mixed dimension In this paper, for subdivision it is proved that max{dim(G), edim(G)} ≤ mdim(S(G)) mdim(G). A family graphs Gn constructed which mdim(Gn) − mdim(S(Gn)) ≥ 2 holds shows inequality can strict, while cactus = For middle dim(M(G)) holds, if tre...

Let G = (V,E) be a graph. A subset D ⊆ V is a total dominating set of G if for every vertex y ∈ V there is a vertex x ∈ D with xy ∈ E. A subset D ⊆ V is a strong dominating set of G if for every vertex y ∈ V − D there is a vertex x ∈ D with xy ∈ E and deg G (x) ≥ deg G (y). The total domination number γt(G) (the strong domination number γS(G)) is defined as the minimum cardinality of a total do...

We study the influence of edge subdivision on the convex domination number. We show that in general an edge subdivision can arbitrarily increase and arbitrarily decrease the convex domination number. We also find some bounds for unicyclic graphs and we investigate graphs G for which the convex domination number changes after subdivision of any edge in G.

 Let $R$ be a commutative ring and $M$ be an $R$-module with $T(M)$ as subset, the set of torsion elements. The total graph of the module denoted by $T(Gamma(M))$, is the (undirected) graph with all elements of $M$ as vertices, and for distinct elements $n,m in M$, the vertices $n$ and $m$ are adjacent if and only if $n+m in T(M)$. In this paper we study the domination number of $T(Gamma(M))$ a...

The preservation of ambient isotopic equivalence under piecewise linear (PL) approximation for smooth knots are prominent in molecular modeling and simulation. Sufficient conditions are given regarding: (1) Hausdorff distance, and (2) a sum of total curvature and derivative. High degree Bézier curves are often used as smooth representations, where computational efficiency is a practical concern...

Subdivision rules have traditionally been designed to generate smooth surfaces from polygonal meshes. In this paper we propose to employ subdivision rules as a polygonal modeling tool, specifically to add additional level of detail to meshes. However, existing subdivision schemes have several undesirable properties making them ill suited for polygonal modeling. In this paper we propose a genera...

The graph sharing game is played by two players, Alice and Bob, on a connected graph G with non-negative weights assigned to the vertices. Starting with Alice, the players take the vertices of G one by one, in each move keeping the set of all taken vertices connected, until the whole G has been taken. Each player wants to maximize the total weight of the vertices they have gathered. It is prove...

Subdivision surface method is a series of iterative operation adopts a certain subdivision formula for an initial grid, obtains the smooth limits surface finally, and can dispose any arbitrary complex topology grid. At present most of the subdivision algorithm are 1-4 subdivisions and as the number of subdivision increase, the grid grow so toorapid in the number of patch that it is difficult fo...