Chapter

# Domination in Chessboards

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## Abstract

This chapter contains a survey of results that have been obtained on domination, independent domination, irredundance, and total domination in chessboard graphs, such as queens, kings, bishops, knights, and rooks graphs.

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A set SV is a dominating set of a graph G = (V;E) if each vertex in V is either in S or is adjacent to a vertex in S. A vertex is said to dominate itself and all its neighbors. The domination number (G) is the minimum cardinality of a dominating set of G. In terms of a chess board problem, let Xn be the graph for chess piece X on the square of side n. Thus, (Xn) is the domination number for chess piece X on the square of side n. In 1964, Yaglom and Yaglom established that (Kn) = � n+2 3 � 2 : This extends to (Km;n) = � m+2 3 �� n+2 3 � for the rectangular board. A set SV is a total dominating set of a graph G = (V;E) if each vertex in V is adjacent to a vertex in S. A vertex is said to dominate its neighbors but not itself. The total domination number t (G) is the minimum cardinality of a total dominating set of G. In 1995, Garnick and Nieuwejaar conducted an analysis of the total domination numbers for the king's graph on the mn board. In this paper we note an error in one portion of their analysis and provide a correct general upperbound for t (Km;n): Furthermore, we state improved upperbounds for t (Kn).
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A set D of vertices of a finite, undirected graph G = (V, E) is a total dominating set if every vertex of V is adjacent to some vertex of D. In this paper we initiate the study of total dominating sets in graphs and, in particular, obtain results concerning the total domination number of G (the smallest number of vertices in a total dominating set) and the total domatic number of G (the largest order of a partition of G into total dominating sets).
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An 1874 article by J. W. L. Glaisher asserted that the eight queens problem of recreational mathematics originated in 1850 with Franz Nauck proposing it to Gauss, who then gave the complete solution. In fact the problem was first proposed two years earlier by Max Bezzel, proposed again by Nauck in a newspaper Gauss happened to read, and only partially solved by Gauss in a casual attempt. Glaisher had access to an accurate account of the history in German but perhaps could not read the language well; the error subsequently spread through the recreational mathematics literature.RésuméEn 1874, J. W. L. Glaisher affirmait dans un article que le problème des huit reines des mathématiques récréatives avait été énoncé pour la première fois par Franz Nauck en 1850 lorsque ce dernier le proposa à Gauss qui en aurait donné alors une solution complète. En fait, Max Bezzel avait déjà proposé ce problème deux années auparavant. Il fut à nouveau énoncé par Nauck dans un journal que Gauss vint à lire. Celui-ci le solutionna partiellement dans une tentative informelle. Glaisher avait accès à une description juste de ces faits en allemand, mais peut-être ne pouvait-il pas lire adéquatement cette langue. Il s'ensuivit que l'erreur se répendit dans toute la littérature sur les mathématiques récréatives.
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The minimum number of queens which can be placed on ann n chessboard so that all other squares are dominated by at least one queen but no queen covers another, is shown to be less than 0.705n + 2.305.
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A vertex x in a subset X of vertices of an undirected graph is redundant if its closed neighbourhood is contained in the union of closed neighbourhoods of vertices of X−{x}. In the context of a communications network, this means that any vertex which may receive communications from X may also be informed from X−{x}. The lower and upper irredundance numbers ir(G) and IR(G) are respectively the minimum and maximum cardinalities taken over all maximal sets of vertices having no redundancies. The domination number γ(G) and upper domination number Γ(G) are respectively the minimum and maximum cardinalities taken over all minimal dominating sets of G. The independent domination number i(G) and the independence number β(G) are respectively the minimum and maximum cardinalities taken over all maximal independent sets of vertices of G.A variety of inequalities involving these quantities are established and sufficient conditions for the equality of the three upper parameters are given. In particular a conjecture of Hoyler and Cockayne [9], namely , is proved.