Tight lower bounds for connected queen domination problems on the chessboard

TL;DR

This paper establishes tight lower bounds for the connected minimum queen domination problem on chessboards, extending to k-colored variants and discussing related extensions.

Contribution

It introduces new lower bounds for connected queen domination, including k-colored variants, and provides insights into extensions of the problem.

Findings

Lower bound of 2N/3-1 for connected queen domination

Extended lower bounds for k-colored variants, decreasing with k

Discussion of problem extensions and future directions

Abstract

1. We first show a lower bound of 2N/3-1 for the connected minimum queen domination (or cover) problem on the NXN chessboard - the upper bound is only 2 higher at most and is easy to show. 2. We then define the k-colored connected minimum queen domination, and extend the above proof to show a lower bound of (2 N - k - 2)/3, where the parameter k can be increased to get decreasing lower bounds LB(N, k) until one reaches the simple domination lower bound of floor N/2. 3. We also discuss extensions of the connected domination problem and additional directions.