Комбінаторна гра — «Зв’язна незв’язність»

Abstract

The combinatorial theory of games is a mathematical theory that examines the games of two persons, where at each moment of time there is a position that players in turn change in a certain way in order to achieve a certain gain. Combination games can be interpreted as games on graphs.In this paper we consider a combinatorial game on a non-oriented graph «Connective incompatibility», which can be used in the simulation of competitive struggle. To solve this problem, an own method of final graphs has been developed, which consists in analyzing the situation that was formed one step before the end of the game. The optimality of the strategy, which results in the complete solution of the problem for an arbitrary number of vertices, is presented in the paper. In the study of the game set the winner, depending on the remainder, this gives the number of vertices when dividing by four.The urgency of this topic is determined by an extremely wide spectrum of the theory of graphs in the modeling of various processes of entrepreneurial activity, etc. The combinatorial theory of games on graphs can be applied in clustering tasks, as well as in the simulation of conflict situations. The difference between combinatorial games from games, which are usually studied in the classical («economic») game theory, is that players play in turns in turn, and not simultaneously (the classical game theory is covered in a multitude of books, which include the words «theory games «or» research operations»).Considerations of the combinatorial theory of games with full information have appeared, even in ancient times, for example, in Sun Tzu's book «The Art of War»: if one can calculate who will win, and not actually fight the war itself.This article can be useful to anyone interested in the combinatorial theory of games, graph theory. The results of this study have different application applications. The topic is promising for further continuation of work in this direction.