A edge 2-rainbow dominating function (E2RDF) of a graph G is a ‎function f from the edge set E(G) to the set of all subsets‎ ‎of the set {1,2} such that for any edge.......................

We prove upper bounds on the order of convergence of lattice based algorithms for numerical integration in function spaces of dominating mixed smoothness on the unit cube with homogeneous boundary condition. More precisely, we study worst-case integration errors for Besov spaces of dominating mixed smoothness B̊p,θ, which also comprise the concept of Sobolev spaces of dominating mixed smoothness...

In this paper, we give a unified approach to error estimates for interpolation on sparse Gauß–Chebyshev grids for multivariate functions from Besov–type spaces with dominating mixed smoothness properties. The error bounds obtained for this method are almost optimal for the considered scale of function spaces. 1991 Mathematics Subject Classification: 41A05, 41A63, 65D05, 46E35

An independent Roman dominating function (IRD-function) on a graph G is f : V ( ) → {0, 1, 2} satisfying the conditions that (i) every vertex u for which = 0 adjacent to at least one v 2, and (ii) set of all vertices assigned non-zero values under independent. The weight an IRD-function sum its over vertices, domination number i R minimum . In this paper, we initiate study bondage b iR having c...

We give a characterization of Sobolev spaces of bivariate periodic functions with dominating smoothness properties in terms of Sobolev spaces of univariate functions. The mixed Sobolev norm is proved to be a uniform crossnorm. This property can be used as a powerful tool in approximation theory. x1. Introduction Beside the approximation of functions from the usual isotropic periodic Sobo-lev sp...

For a simple graph G=(V,E) with no isolated vertices, total Roman {3}-dominating function(TR3DF) on G is function f:V(G)→{0,1,2,3} having the property that (i) ∑w∈N(v)f(w)≥3 if f(v)=0; (ii) ∑w∈N(v)f(w)≥2 f(v)=1; and (iii) every vertex v f(v)≠0 has neighbor u f(u)≠0 for v∈V(G). The weight of TR3DF f sum f(V)=∑v∈V(G)f(v) minimum called {3}-domination number denoted by γt{R3}(G). In this paper, we...

A double Roman dominating function on a graph G=(V,E) is f:V?{0,1,2,3} satisfying the condition that every vertex u for which f(u)=0 adjacent to at least one assigned 3 or two vertices 2, and with f(u)=1 2 3. The weight of f equals w(f)=?v?Vf(v). domination number ?dR(G) G minimum G. We obtain closed expressions generalized Petersen graphs P(5k,k). It proven ?dR(P(5k,k))=8k k?2,3mod5 8k??dR(P(5...

Let k ∈ Z +. A − distance Roman dominating function (kDRDF) on G = (V, E) is a f : V → {0, 1, 2} such that for every vertex v with f(v) 0, there u f(u) 2 d(u, v) ≤ k. The global (GkDRDF) if and only its complement G. weight of the value w(f) P x∈V f(x). minimum graph called domination number denoted as γ gR(G). gR(G) Note that, 1 usual γgR(G), is, γgR(G). authors initiated this study. In paper,...

If G = (V, E, σ) is a finite signed graph, a function f : V → {−1, 0, 1} is a minusdominating function (MDF) of G if f(u) +summation over all vertices v∈N(u) of σ(uv)f(v) ≥ 1 for all u ∈ V . In this paper we characterize signed paths and cycles admitting an MDF.