Appendix to the paper
Bozóki, S., Gál, P., Marosi, I., Weakley, W.D. (2019): Domination of the rectangular queen's graph, The Electronic Journal of Combinatorics 26(4) #P4.45

Sets of 9 queens that dominate the 15×18 chessboard


There are (possibly significantly) more than 4 different solutions; this file contains 4 of them.
Number of Orthodox Cover solutions is: 3.
Number of Centrally strong sets is: 0.
Number of expandable sets is: 2. Maximum area of the expandable sets is: 285. Max exp delta is: 1. Max area is 105% of current board size.

A dominating set is expandable if it covers a larger board than the one given. Expandable sets are noted below, with the possible direction(s) of expansion.

Legend:
Sym (H) denotes reflection symmetry across the horizontal bisector.
Sym (V) denotes reflection symmetry across the vertical bisector.
Sym (C) denotes central symmetry about the center of the board.
Sym (D+) denotes reflection symmetry across the diagonal from lower left corner of the board.
Sym (D-) denotes reflection symmetry across the diagonal from top left corner of the board.
Sym (90) denotes 90 degree rotational symmetry about the center of the board.
The Sym (.) strings are printed without space next to the tables to help searching the solutions.

Δ denotes the absolute value of the difference of the number of queens on dark squares and of the ones on light squares.


15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
123456789101112131415161718

Solution #1

Δ=1
No symmetries
Independent cover

15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
123456789101112131415161718

Solution #2

Δ=9
No symmetries
Orthodox cover

15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
123456789101112131415161718

Solution #3

Δ=5
No symmetries
Orthodox cover
Independent cover
Expandable! Left: 1, Up: 0, Right: 0, Down: 0. Max area: 285

15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
123456789101112131415161718

Solution #4

Δ=5
No symmetries
Orthodox cover
Independent cover
Expandable! Left: 1, Up: 0, Right: 0, Down: 0. Max area: 285