Towards optimal kernel for connected vertex cover in planar graphs

We study the parameterized complexity of the connected version of the vertex cover problem, where the solution set has to induce a connected subgraph. Although this problem does not admit a polynomial kernel for general graphs (unless NP is a subset of coNP/poly), for planar graphs Guo and Niedermeier [ICALP'08] showed a kernel with at most 14k vertices, subsequently improved by Wang et al. [MFCS'11] to 4k. The constant 4 here is so small that a natural question arises: could it be already an optimal value for this problem? In this paper we answer this quesion in negative: we show a (11/3)k-vertex kernel for Connected Vertex Cover in planar graphs. We believe that this result will motivate further study in search for an optimal kernel

PROSUMER ENERGY DIFFUSION DETERMINANTS IN ASSOCIATION WITH LOCAL PLANS OF LOW EMISSION - SMOG - REDUCTION

The cognitive focus of the article concerns the measurement and aggregation of relationships between the diffusion of prosumer energy development and the effectiveness of implementing local Plans of Low Emissions Reduction. In this context, the Authors have justified that local actions in the area of low emission prevention (smog - unusual atmospheric phenomenon) are fully converged with the activities of prosumers, producing energy, among others, in order to meet their own needs. On the basis of the literature query and interviews in a group of managers of the energy and environmental management in territorial units - local perspective (in the selected EU country), levels of energy prosumer transformation have been determined in relation to the measures in the scope of low emission prevention.&nbsp

BIOGAS PLANTS AS AN ELEMENT OF PROSUMER ENERGY DEVELOPMENT IN THE EUROPEAN UNION

The aim of this study is to assess the situation on the market for biogas plants in the EU, an analysis of the environmental benefits of the construction of agricultural biogas plants and the prospects for the development of this industry in the context of the prosumer energy. In the first part of this article the Authors provide basic information about agricultural production of gas in the European Union (total installed capacity, the number of biogas plants, different types of biogas plants, types of substrates used to feed the Biogas plants). The second part of the study provides an overview of the legislation governing prosumer energy and development of this element of the energy market which is an important and permanent part of the EU energy policy. In the third part of this paper the conducted SWOT analysis shows the strengths and weaknesses of the biogas energy, as well as the opportunities and threats of its further development. The scope of the research covers the period 2007-2015. In order to assess biogas plants market development, the Authors have used elements of multivariate statistical analysis and applied Herfindahl-Hirschman Index assessing the degree of concentration in this market

REGIONAL DIVERSIFICATION OF THE LEVEL OF ECONOMIC DEVELOPMENT IN POLAND

In the modern world economy, the importance of regions and cities in economic development is increasing. In line with the policy of the European Union, efforts to level out regional developmental disparities should be made. Regional development can be explored in terms of social, economic or environmental aspects, satisfaction with the level and quality of life of inhabitants, and development of social capital. The aim of this study is to assess only the current level of economic development and its differentiation in various regions of Poland. Data from the Local Data Bank of the Central Statistical Office from the years 2007-2015 have been used in the analysis. Obtained results will allow to draw conclusions about regional policy in Poland