On the metric dimension of corona product graphs

Given a set of vertices $S=\{v_1,v_2,...,v_k\}$ of a connected graph $G$, the metric representation of a vertex $v$ of $G$ with respect to $S$ is the vector $r(v|S)=(d(v,v_1),d(v,v_2),...,d(v,v_k))$, where $d(v,v_i)$, $i\in \{1,...,k\}$ denotes the distance between $v$ and $v_i$. $S$ is a resolving set for $G$ if for every pair of vertices $u,v$ of $G$, $r(u|S)\ne r(v|S)$. The metric dimension of $G$, $dim(G)$, is the minimum cardinality of any resolving set for $G$. Let $G$ and $H$ be two graphs of order $n_1$ and $n_2$, respectively. The corona product $G\odot H$ is defined as the graph obtained from $G$ and $H$ by taking one copy of $G$ and $n_1$ copies of $H$ and joining by an edge each vertex from the $i^{th}$-copy of $H$ with the $i^{th}$-vertex of $G$. For any integer $k\ge 2$, we define the graph $G\odot^k H$ recursively from $G\odot H$ as $G\odot^k H=(G\odot^{k-1} H)\odot H$. We give several results on the metric dimension of $G\odot^k H$. For instance, we show that given two connected graphs $G$ and $H$ of order $n_1\ge 2$ and $n_2\ge 2$, respectively, if the diameter of $H$ is at most two, then $dim(G\odot^k H)=n_1(n_2+1)^{k-1}dim(H)$. Moreover, if $n_2\ge 7$ and the diameter of $H$ is greater than five or $H$ is a cycle graph, then $dim(G\odot^k H)=n_1(n_2+1)^{k-1}dim(K_1\odot H).

Maximal 2-rainbow domination number of a graph

AbstractA 2-rainbow dominating function (2RDF) of a graph G is a function f from the vertex set V(G) to the set of all subsets of the set {1,2} such that for any vertex v∈V(G) with f(v)=0̸ the condition ⋃u∈N(v)f(u)={1,2} is fulfilled, where N(v) is the open neighborhood of v. A maximal 2-rainbow dominating function on a graph G is a 2-rainbow dominating function f such that the set {w∈V(G)|f(w)=0̸} is not a dominating set of G. The weight of a maximal 2RDF f is the value ω(f)=∑v∈V|f(v)|. The maximal 2-rainbow domination number of a graph G, denoted by γmr(G), is the minimum weight of a maximal 2RDF of G. In this paper we initiate the study of maximal 2-rainbow domination number in graphs. We first show that the decision problem is NP-complete even when restricted to bipartite or chordal graphs, and then, we present some sharp bounds for γmr(G). In addition, we determine the maximal rainbow domination number of some graphs

Numerical investigation of influence of the Martensite Volume Fraction on DP steels fracture behavior on the basis of digital material representation model

Development of the methodology for creating reliable digital material representation (DMR) models of dual-phase steels and investigation of influence of the martensite volume fraction on fracture behavior under tensile load are the main goals of the paper. First, an approach based on image processing algorithms for creating a DMR is described. Then, obtained digital microstructures are used as input for the numerical model of deformation, which takes into account mechanisms of ductile fracture. Ferrite and martensite material model parameters are evaluated on the basis of micropillar compression tests. Finally, the model is used to investigate the impact of the martensite volume fraction on the DP steel behavior under plastic deformation. Results of calculations are presented and discussed in the paper

Political risks in project financing at the example of oil transportation system «Odesa–Brody»

Розглянуто причини затримки розвитку проекту ЄАНТК політичного характеру, а також їх вплив на економічну ефективність проекту.There were reviewed the main reasons of delay in Euro-Asian oil transport corridor project that were considered as political by origin. Also the level of their impact on the project was identified

Development of the application of the method for semi-industrial simulation of hot rolling and thermo-mechanical treatment in the research and technological projects

Przedstawiono zakres wykorzystania modułu B linii do półprzemysłowej symulacji procesów wytwarzania wyrobów z metali i stopów (LPS) oraz wyniki badań uzyskane w ramach aktualnie realizowanych i zakończonych w latach 2012–2014 wybranych projektów europejskich i krajowych oraz prac zleconych przez ośrodki badawcze i przedsiębiorców przemysłowych krajowych i zagranicznych. W module B-LPS wykonano fizyczne symulacje walcowania na gorąco wlewków pochodzących z wytopów doświadczalnych i z wsadów z zewnątrz, na pręty i blachy. Walcowanie wyrobów płaskich prowadzono z zastosowaniem regulowanego walcowania z obróbką cieplno-plastyczną i regulowanego chłodzenia po walcowaniu. Materiał po walcowaniu poddano badaniom obejmującym pomiar właściwości mechanicznych i metalograficzną ocenę stanu struktury.This paper presents the range of application of module B of the line for semi-industrial simulation of processes related to manufacturing of metal alloys and products (LPS) as well as results of investigations obtained under selected European and domestic projects in progress and completed in the years 2012–2014 and works commissioned by the research centres and domestic and foreign industrial entrepreneurs. Physical simulations of hot rolling of ingots from experimental melts and external charges for bars and sheets were carried out in the B-LPS module. The rolling of fl at products was carried out using controlled rolling with thermo-mechanical treatment and controlled cooling after rolling. After rolling, the material was subject to investigations including the measurement of mechanical properties and metallographic assessment of structure condition

Experimental verification of carbonation models used for estimation of reinforced concrete structures durability

The subject of the article is a comparison of two types of concrete carbonation models: self-limited carbonation and infinite carbonation. The results of the research on the progress of carbonation during six years of sample exposure in natural atmospheric conditions were used to determine the detailed models for a set of concretes with different w/c and different types of cement, and two scenarios of initial curing. It has been established that the model of self-limiting carbonation (i.e. hyperbolic) is more adequate for describing laboratory tests results in natural conditions