For n=5 and 6 , show that the domination number of the queens graph of an n -by- n chessboard is

For n=5 and 6 , show that the domination number of the...

Key Concepts

Queens Graph

The queens graph is a representation of chessboard squares as vertices of a graph where two vertices are connected if a queen can move from one square to the other in a single move. This graph captures the movement and attack patterns of the queen and provides a framework to study problems related to coverage and control on the chessboard using graph-theoretic methods.

Domination Number

In graph theory, the domination number of a graph is the minimum number of vertices needed such that every vertex in the graph is either one of these vertices or is adjacent to one. It measures how few positions can be used to 'control' or 'dominate' the entire graph, ensuring that every element is influenced or covered by at least one chosen vertex.

Dominating Set

A dominating set in a graph is a set of vertices such that each vertex not in the set is adjacent to at least one vertex in the set. This concept is crucial in optimization problems, where the goal is to find the smallest such set that can exert control over the whole graph, such as placing queens on a chessboard to cover all squares.

Chessboard Coverage and Queen's Moves

Understanding the movement rules of the queen—how it can move along rows, columns, and diagonals—allows one to analyze how a queen placed on a chessboard can attack or cover multiple squares. This geometric insight is fundamental when transforming a chess problem into a graph-theoretical context, where the aim is to efficiently cover every square using the fewest queens possible.

Transcript

0:00 Hello.

00:01 For problem 18, you're asked to solve the eight queens problem.

00:10 So you want to place queens on a chess board so that no two queens are able to attack each other, which means there are no two queens in the same row, no two queens in the same column, and no two queens that are diagonal to each other.

00:28 And so there are lots of ways to go about solving this puzzle.

00:35 I'm just going to present the way that i chose to do it, which is one way of doing it, but know that there are multiple options as to how to do this.

00:45 So we're going to start by setting up our board.

00:48 I'm making a board that is 8 by 8.

00:53 Well, i guess a row is 8.

00:55 And so our board will be 8 by 8, and it's filled with zeros to start.

01:01 So i'm going to work with numbers to start, and then at the end, i'm going to, when i print it out, make sure that it displays it in a way that is similar to the display in the textbook.

01:14 So we want to define a function that checks whether or not there is a queen within attacking distance.

01:23 So define attack from position ij in the board.

01:29 First, we need to check vertically and horizontally to make sure that there are no queens in the same row or columns.

01:36 So for k in range 08, if board ik is equal to 1 or board k, j is equal to 1 return true, because that would mean there is a queen in a row or column, and so you would be able to attack that queen.

01:57 We also need to check diagonally.

01:59 So we'll look for k in range 0 to 8 and l in range 0 to 8, and if k plus l is equal to i plus j or k minus l is equal to i minus j, and if kl is equal to one we want to return true so again we want to look at the diagonals and so if there is a one in one of those diagonal positions then we need to return true otherwise we want to return false so that is our attack function now we want to define queens and this will be the function used to place queens on the board and i'm making this a function of nward n is the number of queens.

02:48 So if n is equal to zero, you're not adding any queens to the board.

02:53 We're just going to return true.

02:56 For i in range 0 to 8 and j in range 0 to 8, we want to place a queen on the board if possible.

03:04 So if attack ij is not true, so it's false...

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