الفهرس 
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Abstract
Game Theory is defined as a means of mathematical analysis when interests collide with each other to reach the best possible decision-making options taking into consideration the given circumstances to get the desired results. Game Theory is related to well-known games such as checkers, XO, and poker. In this dissertation. We will tackle one of the Combinatorial Games in detail, and link another game to the graph, the topic may suggest lots of fun since games are widely recognized as amusing. But the reader should keep in mind that this is a collection of mathematical results and as such has a moderate fun impact.
This dissertation falls into four chapters as following.
Chapter One: In this chapter, we will present introduction to basic concepts of game theory. These include basic definitions of game, classifications of games [4], discussion the Combinatorial Games in detail and explain some examples of these games [7], describing strategic games and Nash Equilibrium, games in extensive and in normal form [5-9], the relationship of game theory to other sciences [9].
Chapter Two: In this chapter, we will discuss in detail the famous game of Tic-Tac-Toe and some different names for this game [10-13], the basic rules for playing, explain some similar games [18], history of the game and how to win this game from the viewpoint of one of the researchers.
Chapter Three: In chapter Three we deduce a new mathematical technique to define the winning game Tic-Tac-Toe. The results were placed in a 3x3 matrix and initial conversions were performed on the rows to find all possible win states. Programming languages were used to find the matrix to determine the diagonal wins. A simulation algorithm is presented to predict the win, or draw of a game by knowing the first move of a player. The game of Tic-Tac-Toe is simulated by using a Min-max algorithm. The results in this chapter have been published in Journal of Game Theory in 2020 entitled “Shahd H. Alkaraz, Essam El-Seidy and Neveen S. Morcos. Tic-Tac-Toe: Understanding the Minimax Algorithm. Journal of Game Theory (2020), 9(1): 1-7.”.
Chapter Four: In this chapter, we provide basic knowledge of definitions and concepts related to the concept of matching in the graph [73]. We are studying a model of games based on two players who take turns adding edges to G, this process eventually produces a maximal matching of the graph. We call the first Maximizer and second player Minimizer. The first aims to get a final matching to be large while the second one wants to reduce it. Maximizer wins if he manages a maximal matching while Minimizer wins if he can prevent him from doing this. The matcher number (α_g ) ́(G) is the number of edges chosen when both players play optimally, while the matching number (α_ ) ́(G) is the number of maximum matching edges. In this chapter we study the relationship between (α_g ) ́(G) and (α_ ) ́(G) .And we also prove some results on types of graph.