How fast can Dominator win in the Maker--Breaker domination game?

We study the Maker--Breaker domination games played by two players, Dominator and Staller. We give a structural characterization for graphs with Maker--Breaker domination number equal to the domination number. Specifically, we show how fast Dominator can win in the game on $P_2 \square P_n$, for $n\geq 1$

Mejker–Brejker igre na grafovima

The topic of this thesis are different variants of Maker–Breaker positional game, where two players Maker and Breaker alternatively take turns in claiming unclaimed edges/vertices of a given graph. We consider Walker–Breaker game, played on the edge set of the graph Kn. Walker, playing the role of Maker is restricted to claim her edges according to a walk, while Breaker can claim any unclaimed edge per move. The focus is on two standard games - the Connectivity game, where Walker has the goal to build a spanning tree on Kn, and the Hamilton Cycle game, where Walker has the goal to build a Hamilton cycle on Kn. We show that Walker with bias 2 can win both games even when playing against Breaker whose bias b is of the order of magnitude n= ln n. Next, we consider (1 : 1) WalkerMaker–WalkerBreaker game on E(Kn),where both Maker and Breaker are walkers and we are interested in seeing how fast WalkerMaker can build spanning tree and Hamilton cycle. Finally, we study Maker–Breaker total domination game played on the vertex set of a given graph. Two players, Dominator and Staller, alternately take turns in claiming unclaimed vertices of the graph. Staller is Maker and wins if she can claim an open neighbourhood of a vertex. Dominator is Breaker and wins if he manages to claim a total dominating set of a graph. For certain connected cubic graphs on n ≥ 6 vertices, we give the characterization of those graphs which are Dominator’s win and those which are Staller’s win.Tema istrazivanja ove disertacije su igre tipa Mejker– Brejker u kojima uˇcestvuju dva igraˇca, Mejker i Brejker, koji naizmjeniˇcno uzimaju slobodne grane/ˇcvorove datog grafa. Bavimo se Voker–Brejker igrama koje se igraju na skupu grana grafa Kn. Voker, u ulozi Mejkera, jeograniˇcen da uzima svoje grane kao da se ˇseta kroz graf, dok Brejker moˇze da uzme bilo koju slobodnu granu grafa. Fokus je na dvije standardne igre - igri povezanosti, gdje Voker ima za cilj da napravi pokrivaju´ce stablo grafa Kn i igri Hamiltonove konture, gdje Voker ima za cilj da napravi Hamiltonovu konturu. Brejker pobjeduje ako sprijeˇci Vokera u ostvarenju njegovog cilja. Pokaza´cemo da Voker sa biasom 2 moˇze da pobijedi u obje igre ˇcak i ako igra protiv Brejkera ˇciji je bias b reda n= ln n. Potom razmatramo (1 : 1) VokerMejker–VokerBrejker igre na Kn, gdje oba igraˇca, i Mejker i Brejker, moraju da biraju grane koje su dio ˇsetnje u njihovom grafu s ciljem odredivanja brze pobjedniˇce strategije VokerMejkera u igri povezanosti i igri Hamiltonove konture. Konaˇcno, istraˇzujemo Mejker–Brejker igre totalne dominacije koje se igraju na skupu ˇcvorova datog grafa. Dva igraˇca, Dom inator i Stoler naizmjeniˇcno uzimaju slobodne ˇcvorove datog grafa. Stoler je Mejker i pobjeduje ako uspije da uzme sve susjede nekog ˇcvora. Dominator je Brejker i pobjeduje ako ˇcvorovi koje uzme dok kraja igre formiraju skup totalne dominacije. Za odredene klase povezanih kubnih grafova reda n ≥ 6, dajemo karakterizaciju onih grafova na kojima Dominator pobjeduje i onih na kojima Stoler pobjeduje.

Maker-Breaker total domination games on cubic graphs

We study Maker-Breaker total domination game played by two players, Dominator and Staller on the connected cubic graphs. Staller (playing the role of Maker) wins if she manages to claim an open neighbourhood of a vertex. Dominator wins otherwise (i.e. if he can claim a total dominating set of a graph). For certain graphs on $n\geq 6$ vertices, we give the characterization on those which are Dominator's win and those which are Staller's win

Mejker–Brejker igre na grafovima

The topic of this thesis are different variants of Maker–Breaker positional game, where two players Maker and Breaker alternatively take turns in claiming unclaimed edges/vertices of a given graph. We consider Walker–Breaker game, played on the edge set of the graph Kn. Walker, playing the role of Maker is restricted to claim her edges according to a walk, while Breaker can claim any unclaimed edge per move. The focus is on two standard games - the Connectivity game, where Walker has the goal to build a spanning tree on Kn, and the Hamilton Cycle game, where Walker has the goal to build a Hamilton cycle on Kn. We show that Walker with bias 2 can win both games even when playing against Breaker whose bias b is of the order of magnitude n= ln n. Next, we consider (1 : 1) WalkerMaker–WalkerBreaker game on E(Kn),where both Maker and Breaker are walkers and we are interested in seeing how fast WalkerMaker can build spanning tree and Hamilton cycle. Finally, we study Maker–Breaker total domination game played on the vertex set of a given graph. Two players, Dominator and Staller, alternately take turns in claiming unclaimed vertices of the graph. Staller is Maker and wins if she can claim an open neighbourhood of a vertex. Dominator is Breaker and wins if he manages to claim a total dominating set of a graph. For certain connected cubic graphs on n ≥ 6 vertices, we give the characterization of those graphs which are Dominator’s win and those which are Staller’s win.Tema istrazivanja ove disertacije su igre tipa Mejker– Brejker u kojima uˇcestvuju dva igraˇca, Mejker i Brejker, koji naizmjeniˇcno uzimaju slobodne grane/ˇcvorove datog grafa. Bavimo se Voker–Brejker igrama koje se igraju na skupu grana grafa Kn. Voker, u ulozi Mejkera, jeograniˇcen da uzima svoje grane kao da se ˇseta kroz graf, dok Brejker moˇze da uzme bilo koju slobodnu granu grafa. Fokus je na dvije standardne igre - igri povezanosti, gdje Voker ima za cilj da napravi pokrivaju´ce stablo grafa Kn i igri Hamiltonove konture, gdje Voker ima za cilj da napravi Hamiltonovu konturu. Brejker pobjeduje ako sprijeˇci Vokera u ostvarenju njegovog cilja. Pokaza´cemo da Voker sa biasom 2 moˇze da pobijedi u obje igre ˇcak i ako igra protiv Brejkera ˇciji je bias b reda n= ln n. Potom razmatramo (1 : 1) VokerMejker–VokerBrejker igre na Kn, gdje oba igraˇca, i Mejker i Brejker, moraju da biraju grane koje su dio ˇsetnje u njihovom grafu s ciljem odredivanja brze pobjedniˇce strategije VokerMejkera u igri povezanosti i igri Hamiltonove konture. Konaˇcno, istraˇzujemo Mejker–Brejker igre totalne dominacije koje se igraju na skupu ˇcvorova datog grafa. Dva igraˇca, Dom inator i Stoler naizmjeniˇcno uzimaju slobodne ˇcvorove datog grafa. Stoler je Mejker i pobjeduje ako uspije da uzme sve susjede nekog ˇcvora. Dominator je Brejker i pobjeduje ako ˇcvorovi koje uzme dok kraja igre formiraju skup totalne dominacije. Za odredene klase povezanih kubnih grafova reda n ≥ 6, dajemo karakterizaciju onih grafova na kojima Dominator pobjeduje i onih na kojima Stoler pobjeduje.

An Overview of Forecasting Methods for Monthly Electricity Consumption

Mid-term electricity consumption forecasting is analysed in this paper. Forecasting of electricity consumption is regression problem that can be defined as using previous consumption of an individual or a group with the goal of calculation of future consumption using some mathematical or statistical approach. The purpose of this prediction is multi beneficial to the stakeholders in the energy community, since this information can affect production, sales and supply. The Different methods are considered with the main goal to determine the best forecasting model. Considered methods include Box-Jenkins autoregressive integrated moving average models, state-space models and exponential smoothing, and machine learning methods including neural networks. An additional objective of the conducted research was to determine if modern methods like machine learning are equally precise in forecasting mid-term electricity consumption when compared to traditional time series methods. The performances of forecasting models are evaluated on the monthly electricity consumption data obtained using real billing software owned by the Distribution System Operator in Bosnia and Herzegovina. Mean absolute percentage error is selected as a measure of prediction accuracy of forecasting methods. Every forecasting method is implemented and tested using the R language, while data is collected from Data Warehouse in the form of total monthly consumption. The efficiency of presented solution will also be discussed after presentation of the results

A Zonal Approach for Wide-Area Temporary Voltage Quality Assessment in a Smart Grid

Wide-area voltage quality assessment represents one of the mandatory objectives for distribution system operators in the development of advanced distribution management systems supporting smart grid requirements. This paper introduces a zonal approach for wide-area temporary voltage quality evaluation in a distribution network. The concept of temporary voltage quality evaluation and assessment is recommended to incentivize active/online management of voltage quality issues. A decision support system based on simple deterministic rules is proposed for rating the voltage quality zones in a distribution network and making recommendations to the distribution system operator. Voltage RMS level, unbalance, and total harmonic distortion are considered voltage quality indices representing the inputs in the decision support system. Residential, commercial, and industrial load types are considered when setting the thresholds for voltage quality indices. The proposed zonal approach for the division of distribution networks in voltage quality zones is applied to the example of a typical European-type distribution network. The operation of a decision support system is tested using the developed distribution smart grid model. The following simulation case studies are conducted: loads with low power factors, manual voltage regulation at MV/LV transformers, unbalanced loads, integration of solar power plant, and nonlinear loads. The obtained simulation results reveal the benefits of the proposed voltage quality assessment approach. Cybersecurity challenges that may impact the proposed approach are addressed, including security vulnerabilities, data privacy, and resilience to cyber threats

Spanning Structures in Walker--Breaker Games

We study the biased $(2:b)$ Walker--Breaker games, played on the edge set ofthe complete graph on $n$ vertices, $K_n$. These games are a variant of theMaker--Breaker games with the restriction that Walker (playing the role ofMaker) has to choose her edges according to a walk. We look at the two standardgraph games -- the Connectivity game and the Hamilton Cycle game and show thatWalker can win both games even when playing against Breaker whose bias is ofthe order of magnitude $n/ \ln n$

An Overview of Forecasting Methods for Monthly Electricity Consumption

Mid-term electricity consumption forecasting is analysed in this paper. Forecasting of electricity consumption is regression problem that can be defined as using previous consumption of an individual or a group with the goal of calculation of future consumption using some mathematical or statistical approach. The purpose of this prediction is multi beneficial to the stakeholders in the energy community, since this information can affect production, sales and supply. The Different methods are considered with the main goal to determine the best forecasting model. Considered methods include Box-Jenkins autoregressive integrated moving average models, state-space models and exponential smoothing, and machine learning methods including neural networks. An additional objective of the conducted research was to determine if modern methods like machine learning are equally precise in forecasting mid-term electricity consumption when compared to traditional time series methods. The performances of forecasting models are evaluated on the monthly electricity consumption data obtained using real billing software owned by the Distribution System Operator in Bosnia and Herzegovina. Mean absolute percentage error is selected as a measure of prediction accuracy of forecasting methods. Every forecasting method is implemented and tested using the R language, while data is collected from Data Warehouse in the form of total monthly consumption. The efficiency of presented solution will also be discussed after presentation of the results