Show that the domination number of the queens graph of an 8 -by-8 chessboary is at most 5 .

Name: Problem 41, Chapter 12: Introductory Combinatorics
Uploaded: 2021-09-13T19:48:14-08:00
Duration: 1 min 58 s
Channel: Sheryl Ezze
Description: Show that the domination number of the queens graph of an 8 -by-8 chessboary is at most 5 .

step 1 We want to look at a sketch of this graph. The k of x is equal to the three secant of two times

Show that the domination number of the queens graph of an 8...

Key Concepts

Proof Technique (Existence via Construction)

A common method to establish an upper bound for a parameter like the domination number is to provide an explicit construction: that is, to show a specific set of vertices (in this case, on the chessboard) that serves as a dominating set. This constructive proof guarantees that the domination number does not exceed the number of vertices in the set provided.

Dominating Set

A dominating set in a graph is a subset of vertices such that every vertex in the graph is either part of this set or adjacent to at least one vertex within the set. Determining a minimal dominating set is a common problem in combinatorial optimization and serves as the basis for the concept of the domination number.

Graph Theory

Graph theory is a field of mathematics that studies structures made up of vertices (or nodes) and edges (connections between nodes). It provides the framework for analyzing and solving problems related to connectivity, covering, and optimization, including domination problems.

Domination Number

The domination number of a graph is the minimum number of vertices needed such that every vertex in the graph is either in this set or adjacent to at least one vertex from the set. It is a measure of how efficiently a few strategic points can 'control' or 'cover' the entire graph.

Queens Graph

The queens graph is a specific graph where the vertices represent the squares of a chessboard, and edges connect two squares if a queen can move from one to the other in a single move (horizontally, vertically, or diagonally). This graph captures the movement and attack patterns of the queen, making it useful in solving domination problems on a chessboard.

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