%0 Journal Article
%T Paired, Total, and Connected Domination on the Queen¡¯s Graph Revisited
%A Paul A. Burchett
%J Open Journal of Discrete Mathematics
%P 1-6
%@ 2161-7643
%D 2016
%I Scientific Research Publishing
%R 10.4236/ojdm.2016.61001
%X 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 <em>n</em> ¡Á <em>n</em> 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 <img src=\"Edit_50aac959-bd38-460d-83e5-f434762c201a.bmp\" alt=\"\" />, <img src=\"Edit_21c145a7-1f1f-4b2f-b7a0-951bac86b509.bmp\" alt=\"\" />, and <img src=\"Edit_c5a8587d-e2d8-45ec-9085-9e8aa67f1e8e.bmp\" alt=\"\" /> respectively. For any <em>n</em> ¡Á <em>n</em> board size, the new bound of <img src=\"Edit_e0b786ac-f2d2-44b8-bd68-7e0ae6fe3cb6.bmp\" alt=\"\" /> is arrived at, along with the separate bounds of <img src=\"Edit_1f061c8f-f191-48f5-9972-a7657fd5e007.bmp\" alt=\"\" /> , for <img src=\"Edit_e723b0c4-b41a-449d-a653-04d525bf41a4.bmp\" alt=\"\" /> with <img src=\"Edit_48a0473a-c9e9-4060-bf94-cba832ded5bf.bmp\" alt=\"\" /> , and <img src=\"Edit_f89174fc-72d9-4523-ba0c-17b2f201dd08.bmp\" alt=\"\" /> , for <img src=\"Edit_ca201372-b17a-430f-ac29-80b71625a72c.bmp\" alt=\"\" /> with <img src=\"Edit_f9d5aedb-1cf4-4348-8509-f4665c2a1deb.bmp\" alt=\"\" />.
%K Chess
%K Total Dominating Set
%K Paired Dominating Set
%K Connected Dominating Set
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=61971