相关论文: Who Wins Domineering on Rectangular Boards?
We consider a minimizing variant of the well-known \emph{No-Three-In-Line Problem}, the \emph{Geometric Dominating Set Problem}: What is the smallest number of points in an $n\times n$~grid such that every grid point lies on a common line…
In a previous article Don Bennett and I looked for,found and proposed a game in which the Standard Model group S(U(2)XU(3)) gets singled out as the "winner". Here I propose to extend this "game" to construct a corresponding game between…
In this paper, we present new upper bounds for the global domination and Roman domination numbers and also prove that these results are asymptotically best possible. Moreover, we give upper bounds for the restrained domination and total…
We examine the complexity of the ``Texas Hold'em'' variant of poker from a topological perspective. We show that there exists a natural simplicial complex governing the multi-way winning probabilities between various hands, and that this…
We investigate a game played between two players, Maker and Breaker, on a countably infinite complete graph where the vertices are the rational numbers. The players alternately claim unclaimed edges. It is Maker's goal to have after…
The domination polynomial of a graph $G$ of order $n$ is the polynomial $D(G,x)=\sum_{i=\gamma(G)}^{n} d(G,i) x^{i}$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$, and $\gamma(G)$ is the domination number of $G$. The…
The aim of this paper is to study the dominant dimension of two important classes of finite dimensional algebras, namely, hereditary algebras and tree algebras. We derive an explicit formula for the dominant dimension of each class.
In this short note, we exhibit a draw in the game of Philosopher's Phutball. We construct a position on a 12 x 10 Phutball board from where either player has a drawing strategy, and then generalize it to an m x n board with m-2 >= n >= 10.
A total dominating set of a graph $G=(V,E)$ is a subset $D$ of $V$ such that every vertex in $V$ is adjacent to at least one vertex in $D$. The total domination number of $G$, denoted by $\gamma _t (G)$, is the minimum cardinality of a…
We study the domination number $\gamma(Q_n^3)$ of the three-dimensional $n \times n \times n$ queen graph. The main result is a stratified theorem computing, for each position type -- corner, edge, face, or interior -- the number of…
We consider tilings of quadriculated regions by dominoes and of triangulated regions by lozenges. We present an overview of results concerning tileability, enumeration and the structure of the space of tilings.
In this paper outer, or dual, billiards outside regular polygons are studied; in particular, periodic points for cases of strictly convex "tables" and for regular n-gons with n = 3,4,6,8,12 are discussed. The main results of the paper are:…
We apply our geometrical theory for counting placements of $q$ nonattacking on an $n\times n$ chessboard, from Parts~I and II, to partial queens: that is, chess pieces with any combination of horizontal, vertical, and $45^\circ$-diagonal…
In this paper, we conclude the calculation of the domination number of all $n\times m$ grid graphs. Indeed, we prove Chang's conjecture saying that for every $16\le n\le m$, $\gamma(G_{n,m})=\lfloor\frac{(n+2)(m+2)}{5}\rfloor -4$.
Abalone is a 2-player board game with perfect information. The game is played on a 5x5x5 hexagonal grid and ends when a player pushes 6 of their opponents' pieces off the board. Abalone is similar to games like chess and Go in that all…
We consider the problem of monitoring an art gallery modeled as a polygon, the edges of which are arcs of curves, with edge or mobile guards. Our focus is on piecewise-convex polygons, i.e., polygons that are locally convex, except possibly…
In a biased weak $(a,b)$ polyform achievement game, the maker and the breaker alternately mark $a,b$ previously unmarked cells on an infinite board, respectively. The maker's goal is to mark a set of cells congruent to a polyform. The…
We introduce a new equivalence relation on the set of all polygonal billiards. We say that two billiards (or polygons) are order equivalent if each of the billiards has an orbit whose footpoints are dense in the boundary and the two…
Player ONE chooses a meager set and player TWO, a nowhere dense set per inning. They play $\omega$ many innings. ONE's consecutive choices must form a (weakly) increasing sequence. TWO wins if the union of the chosen nowhere dense sets…
We introduce a domination polynomial of a graph G. The domination polynomial of a graph G of order n is the polynomial D(G, x) =\sum_{i=1}^n d(G, i)x^i, where d(G, i) is the number of dominating sets of G of size i. We obtain some…