Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Domination Polynomial of the Rook Graph

Stephan Mertens

Abstract

A placement of chess pieces on a chessboard is called dominating, if each free square of the chessboard is under attack by at least one piece. In this contribution we compute the number of dominating arrangements of $k$ rooks on an $n\times m$ chessboard. To this end we derive an expression for the corresponding generating function, the domination polynomial of the $n\times m$ rook graph.

Domination Polynomial of the Rook Graph

Abstract

A placement of chess pieces on a chessboard is called dominating, if each free square of the chessboard is under attack by at least one piece. In this contribution we compute the number of dominating arrangements of rooks on an chessboard. To this end we derive an expression for the corresponding generating function, the domination polynomial of the rook graph.
Paper Structure (5 sections, 5 theorems, 19 equations, 1 figure, 2 tables)

This paper contains 5 sections, 5 theorems, 19 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Recursion
  3. The domination polynomial
  4. Conclusions
  5. Acknowledgments

Key Result

Theorem 1

Let $E_{n,m}(k)$ denote the number of placements of $k$ indistinguishable rooks on an $n\times m$ chessboard such that each row and each column contain at least one rook. Then

Figures (1)

  • Figure 1: 5 queens or 12 knights can dominate the $8\times 8$ board.

Theorems & Definitions (11)

  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Corollary 3
  • proof
  • Theorem 4
  • proof
  • Corollary 5
  • ...and 1 more