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Published online by Cambridge University Press: 29 May 2025
Placement games are a subclass of combinatorial games which are played on graphs. We will demonstrate that one can construct simplicial complexes corresponding to a placement game, and this game could be considered as a game played on these simplicial complexes. These complexes are constructed using square-free monomials.
1. Introduction
We will demonstrate a relationship between a subclass of combinatorial games, such as DOMINEERING and COL, and algebraic structures defined on simplicial complexes. There are two relationships, one via the maximal legal positions and the other through the minimal illegal positions. We will begin by giving the necessary background, first from combinatorial game theory, then from combinatorial commutative algebra.
For a game, perfect information means that both players know which game they are playing, on which board, and the current position. No chance means that no dice can be rolled or cards can be dealt, or any other item involving probability can be used.
Definition 1.1. A combinatorial game is a 2-player game with perfect information and no chance, where the two players are Left and Right (denoted by L and R respectively) and they do not move simultaneously. Then a game is a set P of positions with a specified starting position. Rules determine from which position to which position the players are allowed to move. A legal position is a position that can be reached by playing the game from the starting position (which is legal) according to the rules. Moving from position P to position Q is called a legal move if both P and Q are legal positions and the move is allowed according to the rules. Q is usually called an option of P.
In this paper, a combinatorial game will be denoted by its name in SMALL CAPS. Well-known examples of combinatorial games are CHESS, CHECKERS, TIC-TAC-TOE, GO, and CONNECT FOUR. Examples of games that are not combinatorial games include bridge, backgammon, poker, and snakes and ladders.
Although games usually have a “winning condition” associated to them, i.e., rules as to which player wins, for the purposes of this paper games do not need to have a notion of winning identified.
We will assume that the board on which games are played is a graph (or can be represented as a graph). A space on a board is then equivalent to a vertex and we use the two terms interchangeably.
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