We study the complexity of pathwise approximation of parameter dependent stochastic Itô integration for Cr functions, with r ∈ R, r > 0. Both definite and indefinite integration are considered. This complements previous results [2] for classes of functions with dominating mixed smoothness. Upper bounds are obtained by embedding of function classes and applying some generalizations of these prev...

Dominating set problems are among the most important class of combinatorial in graph optimization, from a theoretical as well practical point view. In this paper, we address recently introduced (minimum) weighted total domination problem. problem, given an undirected with vertex weight function and edge function. The goal is to find dominating D minimal weight. A subset vertices such that every...

<abstract><p>Let $ G be a graph with vertex set V(G) $. A function f:V(G)\rightarrow \{0, 1, 2\} is Roman dominating on if every v\in for which f(v) = 0 adjacent to at least one u\in such that f(u) 2 The domination number of the minimum weight \omega(f) \sum_{x\in V(G)}f(x) among all functions f In this article we study direct product graphs and rooted graphs. Specifically, give sev...

Let G be a graph with no isolated vertex and let N(v) the open neighbourhood of v∈V(G). f:V(G)→{0,1,2} function Vi={v∈V(G):f(v)=i} for every i∈{0,1,2}. We say that f is strongly total Roman dominating on if subgraph induced by V1∪V2 has N(v)∩V2≠∅ v∈V(G)\V2. The domination number G, denoted γtRs(G), defined as minimum weight ω(f)=∑x∈V(G)f(x) among all functions G. This paper devoted to study it ...

Given an undirected graph G = (V,E) and a weight function w : E → R, the Minimum Dominating Tree problem asks to find a minimum weight sub-tree of G, T = (U,F ), such that every v ∈ V \U is adjacent to at least one vertex in U . The special case when the weight function is uniform is known as the Minimum Connected Dominating Set problem. Given an undirected graph G = (V,E) with some subsets of ...

When, as teenagers, we visited the historic site of Vaison-la-Romaine in Southern France, the discovery of Roman capitals buried under the foundations of a mediaeval church was really astounding. How, besides reusing full blocks of the antique architecture, the builders dared reduce the most worked Roman stones to the humble function of backfill? Have new builders become totally disrespectful t...

A two-valued function f defined on the vertices of a graph G = (V,E), f : V → {−1, 1}, is a signed total dominating function if the sum of its function values over any open neighborhood is at least one. That is, for every v ∈ V, f(N(v)) ≥ 1, where N(v) consists of every vertex adjacent to v. The weight of a total signed dominating function is f(V ) = ∑ f(v), over all vertices v ∈ V . The total ...

‎Let G be a graph‎. ‎A 2-rainbow dominating function (or‎ 2-RDF) of G is a function f from V(G)‎ ‎to the set of all subsets of the set {1,2}‎ ‎such that for a vertex v ∈ V (G) with f(v) = ∅, ‎the‎‎condition $bigcup_{uin N_{G}(v)}f(u)={1,2}$ is fulfilled‎, wher NG(v)  is the open neighborhood‎‎of v‎. ‎The weight of 2-RDF f of G is the value‎‎$omega (f):=sum _{vin V(G)}|f(v)|$‎. ‎The 2-rainbow‎‎d...