OPEN Total Even dominating sets: Each vertex must have even number of neighbors, not counting itself. This requires a NON-ZERO number of neighbors. These tend to have pretty patterns such as 11101111111111111111 10101000000000000001 11101011111111111101 00001010000000000101 11111010111111110101 10000010100000010101 10111110101111010101 10100000101001010101 10101111101001010101 10101000001111010101 10101011110000010101 10101010010111110101 10101010010100000101 10101011110101111101 10101000000101000001 10101111111101011111 10100000000001010000 10111111111111010111 10000000000000010101 11111111111111110111 The following grid dimensions have these: 2 by 2 3 by 3 4 by 4 5 by 2 6 by 6 7 by {3, 7} 8 by {2, 8} 9 by {4} 10 by 10 11 by {2, 3, 11} 12 by 12 13 by {6} 14 by {2, 4, 14} 15 by {3, 7, 15} 16 by 16 17 by {2, 8} 18 by 18 19 by {3, 4, 19] 20 by {2, 6, 20} Total Even dominating sets: Each vertex must have even number of neighbors, counting itself. This requires a NON-ZERO number of neighbors. The following grid dimensions have these: 2 by {1, 3} 3 by {2, 5} 4 by 4 5 by {1, 2, 3, 5, 7, 11} 6 by 8 7 by {2, 5, 11} 8 by {1, 6, 13} 9 by {2, 4, 14} 11 by {1, 2, 3, 5, 7, 11, 15} 13 by {2, 8} 14 by {1, 4, 9, 16} 15 by {2, 11, 23} 16 by 14 17 by {1, 2, 13} 18 none 19 by {2, 4}