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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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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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.