OPEN Total Odd dominating sets:

Each vertex must have odd number of neighbors,
not counting itself. 

This is the same as open odd dominating set.


Total Odd dominating sets:

Each vertex must have odd number of neighbors,
counting itself. A vertex in the set must have
at least 3 neighbors in the set (counting itself).

The following grids have these:

2 by {1, 2}

3 by {2,  3}

4 by {2, 3, 4}

5 by {3, 4}

6 by {2, 3, 5, 6}

7 by {2, 6, 7}

8 by {2, 4, 6, 8}

9 by {4, 6}

10 by {2, 3, 4, 10}

11 by {2, 3}

12 by {2, 3, 12}

13 by {4}

14 by {2, 4, 8}

15 by {2, 4}

16 by {2, 3, 6}

These seem exceedingly rare (in the cases where they
do exist, e.g. 14 by 8, there are not many of them).