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Games and Complexes I: Transformation via Ideals
Abstract and Figures
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.
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... In [3] it was shown that a placement game G played on a board B is equivalent to a simplicial complex ∆ G,B . We look at weight games, a subclass of placement games, and introduce upper bounds on the number of positions with i pieces in G, or equivalently the number of faces with i vertices in ∆ G,B , which are reminiscent of the Kruskal-Katona bounds. ...
... In [3] we demonstrated that to a placement game G played on a board B one can associate a simplicial complex ∆ G,B where G can be considered as a game played on ∆ G,B . ...
... We begin by introducing some of the concepts needed. A complete introduction is given in [3]. Definition 1.1 (Brown et al. [2]). ...
In [3] it was shown that a placement game G played on a board B is equivalent
to a simplicial complex \Delta_{G,B}. We look at weight games, a subclass of
placement games, and introduce upper bounds on the number of positions with i
pieces in G, or equivalently the number of faces with i vertices in
\Delta_{G,B}, which are reminiscent of the Kruskal-Katona bounds.
... In [12] we initiated the idea of using simplicial complexes to algebraically describe SP-games, a class of combinatorial games. To each SP-game we can assign two simplicial complexes, one representing all legal positions, the so called legal complex, and one representing the minimal illegal positions, the illegal complex. ...
... This example also illustrates the following result. See [12] for more details. ...
Strong placement games (SP-games) are a class of combinatorial games whose structure allows one to describe the game via simplicial complexes. A natural question is whether well-known invariants of combinatorial games, such as "game value", appear as invariants of the simplicial complexes. This paper is the first step in that direction. We show that every simplicial complex encodes a certain type of SP-game (called an "invariant SP-game") whose ruleset is independent of the board it is played on. We also show that in the class of SP-games isomorphic simplicial complexes correspond to isomorphic game trees, and hence equal game values. We also study a subclass of SP-games corresponding to flag complexes, showing that there is always a game whose corresponding complex is a flag complex no matter which board it is played on.
... Placement games that satisfy the additional constraint, that if a position can be reached by a sequence of legal moves, then any sequence of moves which reach this position is legal, are called strong placement games. This class of games was introduced by Faridi, Huntemann, and Nowakowski in [15], while the theory of strong placement games has been expanded upon in [16,23,24,26]. Notice that Digraph placement is not a strong placement ruleset. ...
... For a formal statement of this see Theorem 3.1. This result is a natural analogue to Theorem 4.4 from [15] which shows that every strong placement game is an instance of a position in a given ruleset played on a simplicial complex. ...
This paper considers a natural ruleset for playing a partisan combinatorial game on a directed graph, which we call Digraph Placement. Given a digraph G with a not necessarily proper 2-coloring of V(G), the Digraph Placement game played on G by the players Left and Right, who play alternately, is defined as follows. On her turn, Left chooses a blue vertex which is deleted along with all of its out-neighbours. On his turn Right chooses a red vertex, which is deleted along with all of its out-neighbours. A player loses if on their turn they cannot move. We show constructively that Digraph Placement is a universal partisan ruleset; for all partisan combinatorial games X there exists a Digraph Placement game, G, such that . Digraph Placement and many other games including Nim, Poset Game, Col, Node Kayles, Domineering, and Arc Kayles are instances of a class of placement games that we call conflict placement games. We prove that X is a conflict placement game if and only if it has the same literal form as a Digraph Placement game. A corollary of this is that deciding the winner of a Digraph Placement game is PSPACE-hard. Next, for a game value X we prove bounds on the order of a smallest Digraph Placement game G such that .
... Domineering belongs to the class of strong placement games, and thus one can assign to each game a simplicial complex representing the legal positions (see [5] for details). The coefficients of D m,n (x, x) are then the entries of the f -vectorthe vector counting the number of faces of a given dimension -of this simplicial complex, while the coefficients of F m,n (x, x) give the number of maximal faces, called facets, of a fixed dimension. ...
Domineering is a two player game played on a checkerboard in which one player places dominoes vertically and the other places them horizontally. We give bivariate generating polynomials enumerating Domineering positions by the number of each player's pieces. We enumerate all positions, maximal positions, and positions where one player has no move. Using these polynomials we count the number of positions that occur during alternating play. Our method extends to enumerating positions from mid-game positions and we include an analysis of a tournament game.
... In [9] we initiated the idea of using simplicial complexes to algebraically describe SP-games, a class of combinatorial games. To each SP-game we can assign two simplicial complexes, one representing all legal positions, the so called legal complex, and one representing the minimal illegal positions, the illegal complex. ...
Strong placement games (SP-games) are a class of combinatorial games whose structure allows one to describe the game via simplicial complexes. A natural question is whether well-known invariants of combinatorial games, such as "game value", appear as invariants of the simplicial complexes. This paper is the first step in that direction. We show that every simplicial complex encodes a certain type of SP-game (called an "invariant SP-game") whose ruleset is independent of the board it is played on. We also show that in the class of SP-games isomorphic simplicial complexes correspond to isomorphic game trees, and hence equal game values. We also study a subclass of SP-games corresponding to flag complexes, showing that there is always a game whose corresponding complex is a flag complex no matter which board it is played on.
... Distance games were introduced by Huntemann and Nowakowski [8]. They are part of a larger class of combinatorial games called placement games studied in [3] and [6]. Distance games are played on a graph (board). ...
We study the computational complexity of distance games, a class of combinatorial games played on graphs. A move consists of colouring an uncoloured vertex subject to it not being at certain distances determined by two sets, D and S. D is the set of forbidden distances for colouring vertices in different colors, while S is the set of forbidden distances for the same colour. The last player to move wins. Well-known examples of distance games are Node-Kayles, Snort, and Col, whose complexities were shown to be PSPACE-hard. We show that many more distance games are also PSPACE-hard.
This paper addresses several significant gaps in the theory of restricted mis\`ere play (Plambeck, Siegel 2008), primarily in the well-studied universe of dead-ending games, (Milley, Renault 2013); if a player run out of moves in , then they can never move again in any follower of X. A universe of games is a class of games which is closed under disjunctive sum, taking options and conjugates. We use novel results from absolute combinatorial game theory (Larsson, Nowakowski, Santos 2017) to show that and the universe of dicot games (either both, or none of the players can move) have 'options only' test for comparison of games, and this in turn is used to define unique reduced games (canonical forms) in . We develop the reductions for by extending analogous work for , in particular by solving the problem of reversibility through ends in the larger universe. Finally, by using the defined canonical forms in and , we prove that both of these universes, as well as the subuniverse of impartial games, have the conjugate property: every inverse game is obtained by swapping sides of the players.
On considère généralement que la théorie des graphes est née au 18e siècle, et qu'elle connaît un essor significatif depuis les années 1960. L'avènement de la théorie des jeux combinatoires est quant à lui plus récent (fin des années 1970). Ce domaine reste alors moins exploré dans la littérature, et de nombreuses études sur des techniques générales de résolution sont toujours actuellement en cours de construction. Dans ce mémoire, je propose plusieurs tours d'horizons à propos de problématiques bien ciblées de ces deux domaines.Dans un premier temps, je m'interroge sur la complexité des règles de jeux de suppression de tas. Il s'avère que dans la littérature, la complexité d'un jeu est souvent définie comme la complexité algorithmique d'une stratégie gagnante. Cependant, il peut aussi avoir du sens de s'interroger sur la nature des règles de jeu. Un premier pas dans cette direction a été fait avec l'introduction du concept de jeu dit invariant. On notera au passage que certains résultats obtenus ont mis en exergue des liens entre combinatoire des mots et stratégie gagnante d'un jeu. Dans un deuxième chapitre, j'aborde les jeux sous l'angle des graphes. Deux aspects sont considérés:* Un graphe peut être vu comme un support de jeu. Le cas du jeu de Nim et ses variantes sur les graphes y est examiné.* Certaines problématiques standard de théorie des graphes peuvent être transformées dans une version ludique. C'est d'ailleurs un objet d'étude de plus en plus prisé par la communauté. Nous détaillerons le cas des jeux de coloration sommet.Enfin, le dernier chapitre se concentre sur deux nouvelles variantes de problématiques issues de la théorie des graphes: le placement de graphes et les colorations distinguantes. J'en profite pour faire un état de l'art des principaux résultats sur ces deux domaines.
In [3] it was shown that a placement game G played on a board B is equivalent
to a simplicial complex \Delta_{G,B}. We look at weight games, a subclass of
placement games, and introduce upper bounds on the number of positions with i
pieces in G, or equivalently the number of faces with i vertices in
\Delta_{G,B}, which are reminiscent of the Kruskal-Katona bounds.
We present several results concerning the number of positions and games of Go. We derive recurrences for L(m,n), the number of legal positions on an m ×n board, and develop a dynamic programming algorithm which computes L(m,n) in time O(m
3n
2λ
m
) and space O(m
λ
m
), for some constant λ< 5.4. An implementation of this algorithm enables us to list L(n,n) for n ≤ 17. For larger boards, we prove the existence of a base of liberties
of ~2.9757341920433572493. Based on a conjecture about vanishing error terms, we derive an asymptotic formula for L(m,n), which is shown to be highly accurate.
We also study the Game Tree complexity of Go, proving an upper bound on the number of possible games of (mn)L(m,n) and a lower bound of on n×nboards and on 1 ×n boards, in addition to exact counts for mn ≤ 4 and estimates up to mn = 9. Most proofs and some additional results had to be left out to observe the page limit. They may be found in the full version available at [8].
In the last two decades Cohen-Macaulay rings and modules have been central topics in commutative algebra. This book meets the need for a thorough, self-contained introduction to the homological and combinatorial aspects of the theory of Cohen-Macaulay rings, Gorenstein rings, local cohomology, and canonical modules. A separate chapter is devoted to Hilbert functions (including Macaulay's theorem) and numerical invariants derived from them. The authors emphasize the study of explicit, specific rings, making the presentation as concrete as possible. So the general theory is applied to Stanley-Reisner rings, semigroup rings, determinantal rings, and rings of invariants. Their connections with combinatorics are highlighted, e.g. Stanley's upper bound theorem or Ehrhart's reciprocity law for rational polytopes. The final chapters are devoted to Hochster's theorem on big Cohen-Macaulay modules and its applications, including Peskine-Szpiro's intersection theorem, the Evans-Griffith syzygy theorem, bounds for Bass numbers, and tight closure. Throughout each chapter the authors have supplied many examples and exercises which, combined with the expository style, will make the book very useful for graduate courses in algebra. As the only modern, broad account of the subject it will be essential reading for researchers in commutative algebra.
This paper introduces graph polynomials based on a concept from the game of Go. Suppose that, for each vertex of a graph, we either leave it uncoloured or choose a colour uniformly at random from a set of available colours, with the choices for the vertices being independent and identically distributed. We ask for the probability that the resulting partial assignment of colours has the following property: for every colour class, each component of the subgraph it induces has a vertex that is adjacent to an uncoloured vertex. In Go terms, we are requiring that every group is uncaptured. This definition leads to Go polynomials for a graph. Although these polynomials are based on properties that are less “local” in nature than those used to define more traditional graph polynomials such as the chromatic polynomial, we show that they satisfy recursive relations based on local modifications similar in spirit to the deletion–contraction relation for the chromatic polynomial. We then show that they are #P-hard to compute in general, using a result on linear forms in logarithms from transcendental number theory. We also briefly record some correlation inequalities.
We use transfer matrix methods to determine bounds for the numbers of legal Go positions for various numbers of players on some planar lattice graphs, including square lattice graphs such as those on which the game is normally played. We also Þnd bounds on limiting constants that describe the behaviour of the number of legal positions on these lattice graphs as the dimensions of the lattices tend to inÞnity. These results amount to giving bounds for some speciÞc evaluations of Go polynomials on these graphs.
To a simplicial complex, we associate a square-free monomial ideal in the polynomial ring generated by its vertex set over a field. We study algebraic properties of this ideal via combinatorial properties of the simplicial complex. By generalizing the notion of a tree from graphs to simplicial complexes, we show that ideals associated to trees satisfy sliding depth condition, and therefore have normal and Cohen-Macaulay Rees rings. We also discuss connections with the theory of Stanley-Reisner rings.
Combinatorics of go Dalhousie University, Halifax, Nova Scotia, B3H 3J5, Canada E-mail address: faridi@mathstat.dal.ca, svenjah@mathstat.dal.ca, rjn@mathstat
- J Tromp
- G Farnebäck
J. Tromp and G. Farnebäck. Combinatorics of go. In Proceedings of 5th International Conference on Computer and Games, Torino, Italy, May 2006. Dalhousie University, Halifax, Nova Scotia, B3H 3J5, Canada E-mail address: faridi@mathstat.dal.ca, svenjah@mathstat.dal.ca, rjn@mathstat.dal.ca