The question associated with total domination on the queen's graph has a long and rich history, first having been posed by Ahrens in 1910 [1]. The question is this: What is the minimum number of queens needed so that every square of an n × n board is attacked? Beginning in 2005 with Amirabadi, Burchett, and Hedetniemi [2] [3], work on this problem, and two other related problems, has seen progress. Bounds have been given for the values of all three domination parameters on the queen's graph. In this paper, formations of queens are given that provide new bounds for the values of total, paired, and connected domination on the queen's graph, denoted

In this paper the theological views of both Richard Dawkins and Bertrand Russell are refined to make them more precise and consistent. Russell points famously to an example of a celestial teacup as an analogy for the existence of God. More specifically the analogy is used to argue against attempts to shift the burden of proof against the sceptic. In his book The God Delusion, Dawkins picks up on this example and uses it as a foundation to argue against agnosticism. While Dawkins' larger framework for considering the existence of God may or may not be appropriate, it is important that the refinements are emphasized to make the arguments clear. In short his "Spectrum of Theistic Probabilities", as introduced in his book The God Delusion, is adapted to include the consideration of infinite sets. This consideration makes Dawkins' own expressions of his viewpoint, and the teacup analogy more consistent-as Dawkins' categorization of his own views seem to push the use of the spectrum to its limit. For example, he claims his views don't fall precisely onto one of the 7 milestones of his spectrum. The author offers a correction of this by categorizing Dawkins' own views neatly onto one of the refined milestones.

For a chessboard graph and a given graph parameter π, a π separation number is the minimum number of pawns for which some arrangement of those pawns on the board will produce a board where π has some desired value. We extend previous results on independence and domination separation. We also consider separation of other domination-related parameters. c

In this paper, both the roman domination number and the number of minimum roman dominating sets are found for any rectangular rook's graph. In a similar fashion, the roman domination number and the number of minimum roman dominating sets are found on the square bishop's graph for odd board sizes. Also found are the number of minimum total dominating sets associated with the light-colored squares when 7 mod12 n ≡ .

Both independence and independence-separation problems on chessboard graphs have been studied in detail, with hundreds of papers in the broader independence category, and several on the independence-separation problem variant for chessboard graphs. In this paper, the independence-separation problem is considered on the d-dimensional rook's graph. A lower bound of