Domination and Independence on the Rectangular Torus by Rooks and Bishops
نویسندگان
چکیده
A set S V 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. A set S V is an independent set of vertices if no two vertices in S are adjacent. The independence number, B0 (G), is the maximum cardinality of an independent set of G. Both (G) and B0 (G) are pieces of the six part domination chain: ir (G) (G) i (G) B0 (G) (G) IR (G). Watkins has computed the domination numbers of rooks and bishops on the square torus. In this paper we compute the domination, total domination, independent domination and independence numbers of the bishop and rook on the rectangularm n toroidal board.
منابع مشابه
Queens graphs for chessboards on the torus
Ve consider the independence, domination and independent domination numbers of graphs obtained from the moves of queens on chessboards drawn on the torus, and determine exact values for each of these parameters in infinitely many cases.
متن کاملMixed Roman domination and 2-independence in trees
Let $G=(V, E)$ be a simple graph with vertex set $V$ and edge set $E$. A {em mixed Roman dominating function} (MRDF) of $G$ is a function $f:Vcup Erightarrow {0,1,2}$ satisfying the condition that every element $xin Vcup E$ for which $f(x)=0$ is adjacentor incident to at least one element $yin Vcup E$ for which $f(y)=2$. The weight of anMRDF $f$ is $sum _{xin Vcup E} f(x)$. The mi...
متن کاملA q-QUEENS PROBLEM IV. QUEENS, BISHOPS, NIGHTRIDERS (AND ROOKS)
Parts I–III showed that the number of ways to place q nonattacking queens or similar chess pieces on an n × n chessboard is a quasipolynomial function of n whose coefficients are essentially polynomials in q and, for pieces with some of the queen’s moves, proved formulas for these counting quasipolynomials for small numbers of pieces and highorder coefficients of the general counting quasipolyn...
متن کاملI.R. and Independence-Seeking Based in the Ideal Sky and Civilized Horizon
In this study the impact of the Islamic revolution on freedom as an important achievement, and the changes and levels of independence in the growing process of the revolution. In a negative approach, the research proposes the hypothesis that the Islamic Revolution has opposed the Westphalian state-based independence and developed Islamic teaching-based independence based. The present study, wit...
متن کاملThe Rule of "Nafye Sabil" [i.e. to prevent the Islamic society to be dominated by non-Muslims] in Islamic Thought and Foreign Policy of Islamic Republic of Iran
As a jurisprudential rule, "nafye sabil" has played a sustaining and influential role in Islamic system's major decisions, policies and behavior. This principle is of high importance in Islamic state's foreign relations. Rejecting oppression and tyranny against Muslims and preserving their freedom and removing dependence on aliens is the foundation of this rule in Islamic republic of Iran's for...
متن کامل