Open Odd Dominating Sets in Grids (even vertex v must have |N(v) intersect D| odd. For a given number of columns n, we generate n by m grids having open even dominating sets. This is done by considering an 2^n possible values D for the first row of the grid. Iteration continued until 3rd row of zeroes 0 1 0 o o 1 x x 2 o o 3 o o 4 x x 5 o o 0 1 0 o x 1 o x 2 o o 3 x o 4 x o 5 o o 6 o x 7 o x 8 o o 0 1 0 x o 1 x o 2 o o 3 o x 4 o x 5 o o 6 x o 7 x o 8 o o 0 1 0 x x 1 o o 2 o o 3 x x 4 o o Total Time: 0.00 seconds Graphs of size 2: Number with max value (8): 2 Percentage with max value: 50.000% FREQUENCY TABLE [value : frequency] 4: 1 5: 1 8: 2 *** Line of 0's terminated after 5 lines ============================================= 0 1 2 0 o o o 1 x x x 2 o x o 3 x o x 4 x o x 5 o x o 6 x x x 7 o o o 8 o o o 9 x x x 10 o x o 11 x o x 12 x o x 13 o x o 14 x x x 15 o o o 0 1 2 0 o o x 1 x o x 2 x x o 3 x o x 4 o o x 5 o o o 6 x x o 7 o o o 8 o o x 9 x o x 10 x x o 11 x o x 12 o o x 13 o o o 0 1 2 0 o x o 1 o x o 2 o o o 3 x o x 4 x x x 5 x x x 6 x o x 7 o o o 8 o x o 9 o x o 10 o o o 0 1 2 0 o x x 1 o o o 2 x o o 3 x o x 4 o x x 5 x o x 6 x o o 7 o o o 8 o x x 9 o o o 0 1 2 0 x o o 1 x o x 2 o x x 3 x o x 4 x o o 5 o o o 6 o x x 7 o o o 8 x o o 9 x o x 10 o x x 11 x o x 12 x o o 13 o o o 0 1 2 0 x o x 1 x x x 2 x x x 3 x o x 4 o o o 5 o x o 6 o x o 7 o o o 8 x o x 9 x x x 10 x x x 11 x o x 12 o o o 0 1 2 0 x x o 1 o o o 2 o o x 3 x o x 4 x x o 5 x o x 6 o o x 7 o o o 8 x x o 9 o o o 0 1 2 0 x x x 1 o x o 2 x o x 3 x o x 4 o x o 5 x x x 6 o o o 7 o o o 8 x x x 9 o x o 10 x o x 11 x o x 12 o x o 13 x x x 14 o o o Total Time: 0.00 seconds Graphs of size 3: Number with max value (15): 1 Percentage with max value: 12.500% FREQUENCY TABLE [value : frequency] 9: 2 10: 1 12: 1 13: 2 14: 1 15: 1 *** Line of 0's terminated after 15 lines ============================================= 0 1 2 3 0 o o o o 1 x x x x 2 o x x o 3 x x x x 4 o o o o 5 o o o o 6 x x x x 7 o x x o 8 x x x x 9 o o o o 0 1 2 3 0 o o o x 1 x x o x 2 o o x o 3 o x x x 4 o o o o 5 x o o o 6 x o x x 7 o x o o 8 x x x o 9 o o o o 10 o o o x 11 x x o x 12 o o x o 13 o x x x 14 o o o o 0 1 2 3 0 o o x o 1 x o x o 2 x x o o 3 x o x x 4 o o o o 5 o x o o 6 o x o x 7 o o x x 8 x x o x 9 o o o o 10 o o x o 11 x o x o 12 x x o o 13 x o x x 14 o o o o 0 1 2 3 0 o o x x 1 x o o o 2 x o o o 3 o o x x 4 o o o o 5 x x o o 6 o o o x 7 o o o x 8 x x o o 9 o o o o 10 o o x x 11 x o o o 12 x o o o 13 o o x x 14 o o o o 0 1 2 3 0 o x o o 1 o x o x 2 o o x x 3 x x o x 4 o o o o 5 o o x o 6 x o x o 7 x x o o 8 x o x x 9 o o o o 10 o x o o 11 o x o x 12 o o x x 13 x x o x 14 o o o o 0 1 2 3 0 o x o x 1 o x x x 2 o x x x 3 o x o x 4 o o o o 5 x o x o 6 x x x o 7 x x x o 8 x o x o 9 o o o o 10 o x o x 11 o x x x 12 o x x x 13 o x o x 14 o o o o 0 1 2 3 0 o x x o 1 o o o o 2 x o o x 3 x o o x 4 o o o o 5 o x x o 6 o o o o 0 1 2 3 0 o x x x 1 o o x o 2 x x o x 3 o o o x 4 o o o o 5 x x x o 6 o x o o 7 x o x x 8 x o o o 9 o o o o 10 o x x x 11 o o x o 12 x x o x 13 o o o x 14 o o o o 0 1 2 3 0 x o o o 1 x o x x 2 o x o o 3 x x x o 4 o o o o 5 o o o x 6 x x o x 7 o o x o 8 o x x x 9 o o o o 10 x o o o 11 x o x x 12 o x o o 13 x x x o 14 o o o o 0 1 2 3 0 x o o x 1 x o o x 2 o o o o 3 o x x o 4 o o o o 5 x o o x 6 x o o x 7 o o o o 0 1 2 3 0 x o x o 1 x x x o 2 x x x o 3 x o x o 4 o o o o 5 o x o x 6 o x x x 7 o x x x 8 o x o x 9 o o o o 10 x o x o 11 x x x o 12 x x x o 13 x o x o 14 o o o o 0 1 2 3 0 x o x x 1 x x o o 2 x o x o 3 o o x o 4 o o o o 5 x x o x 6 o o x x 7 o x o x 8 o x o o 9 o o o o 10 x o x x 11 x x o o 12 x o x o 13 o o x o 14 o o o o 0 1 2 3 0 x x o o 1 o o o x 2 o o o x 3 x x o o 4 o o o o 5 o o x x 6 x o o o 7 x o o o 8 o o x x 9 o o o o 10 x x o o 11 o o o x 12 o o o x 13 x x o o 14 o o o o 0 1 2 3 0 x x o x 1 o o x x 2 o x o x 3 o x o o 4 o o o o 5 x o x x 6 x x o o 7 x o x o 8 o o x o 9 o o o o 10 x x o x 11 o o x x 12 o x o x 13 o x o o 14 o o o o 0 1 2 3 0 x x x o 1 o x o o 2 x o x x 3 x o o o 4 o o o o 5 o x x x 6 o o x o 7 x x o x 8 o o o x 9 o o o o 10 x x x o 11 o x o o 12 x o x x 13 x o o o 14 o o o o 0 1 2 3 0 x x x x 1 o x x o 2 x x x x 3 o o o o 4 o o o o 5 x x x x 6 o x x o 7 x x x x 8 o o o o Total Time: 0.01 seconds Graphs of size 4: Number with max value (14): 12 Percentage with max value: 75.000% FREQUENCY TABLE [value : frequency] 6: 1 7: 1 8: 1 9: 1 14: 12 *** Line of 0's terminated after 9 lines ============================================= Total Time: 0.00 seconds Graphs of size 5: Number with max value (35): 16 Percentage with max value: 50.000% FREQUENCY TABLE [value : frequency] 14: 4 15: 2 16: 1 18: 1 19: 2 20: 4 22: 1 23: 1 35: 16 *** Line of 0's terminated after 23 lines ============================================= Total Time: 0.00 seconds Graphs of size 6: Number with max value (20): 58 Percentage with max value: 90.625% FREQUENCY TABLE [value : frequency] 8: 1 9: 1 10: 1 11: 1 12: 1 13: 1 20: 58 *** Line of 0's terminated after 13 lines ============================================= Total Time: 0.01 seconds Graphs of size 7: Number with max value (47): 96 Percentage with max value: 75.000% FREQUENCY TABLE [value : frequency] 17: 2 18: 1 19: 8 20: 1 21: 2 22: 1 24: 1 25: 2 26: 1 27: 8 28: 1 29: 2 30: 1 31: 1 47: 96 *** Line of 0's terminated after 31 lines ============================================= Total Time: 0.01 seconds Graphs of size 8: Number with max value (26): 242 Percentage with max value: 94.531% FREQUENCY TABLE [value : frequency] 10: 1 11: 4 12: 1 13: 1 14: 4 15: 1 16: 1 17: 1 26: 242 *** Line of 0's terminated after 17 lines ============================================= Total Time: 0.05 seconds Graphs of size 9: Number with max value (59): 464 Percentage with max value: 90.625% FREQUENCY TABLE [value : frequency] 22: 1 23: 2 24: 16 26: 1 27: 2 28: 1 30: 1 31: 2 32: 1 34: 16 35: 2 36: 1 38: 1 39: 1 59: 464 *** Line of 0's terminated after 39 lines ============================================= Total Time: 0.06 seconds Graphs of size 10: Number with max value (32): 1014 Percentage with max value: 99.023% FREQUENCY TABLE [value : frequency] 12: 1 13: 1 14: 1 15: 1 16: 1 17: 1 18: 1 19: 1 20: 1 21: 1 32: 1014 *** Line of 0's terminated after 21 lines ============================================= Total Time: 0.27 seconds Graphs of size 11: Number with max value (71): 1936 Percentage with max value: 94.531% FREQUENCY TABLE [value : frequency] 25: 2 26: 4 28: 1 29: 32 30: 1 31: 8 32: 4 33: 2 34: 1 36: 1 37: 2 38: 4 39: 8 40: 1 41: 32 42: 1 44: 4 45: 2 46: 1 47: 1 71: 1936 *** Line of 0's terminated after 47 lines ============================================= Total Time: 0.33 seconds Graphs of size 12: Number with max value (38): 4084 Percentage with max value: 99.707% FREQUENCY TABLE [value : frequency] 14: 1 15: 1 16: 1 17: 1 18: 1 19: 1 20: 1 21: 1 22: 1 23: 1 24: 1 25: 1 38: 4084 *** Line of 0's terminated after 25 lines ============================================= Total Time: 1.50 seconds Graphs of size 13: Number with max value (83): 8040 Percentage with max value: 98.145% FREQUENCY TABLE [value : frequency] 30: 1 31: 2 32: 1 34: 64 35: 2 36: 1 38: 1 39: 2 40: 1 42: 1 43: 2 44: 1 46: 1 47: 2 48: 64 50: 1 51: 2 52: 1 54: 1 55: 1 83: 8040 *** Line of 0's terminated after 55 lines ============================================= Total Time: 1.74 seconds Graphs of size 14: Number with max value (44): 16328 Percentage with max value: 99.658% FREQUENCY TABLE [value : frequency] 16: 1 17: 4 18: 1 19: 16 20: 4 21: 1 22: 1 23: 4 24: 16 25: 1 26: 4 27: 1 28: 1 29: 1 44: 16328 *** Line of 0's terminated after 29 lines ============================================= Total Time: 7.81 seconds Graphs of size 15: Number with max value (95): 32448 Percentage with max value: 99.023% FREQUENCY TABLE [value : frequency] 33: 2 34: 1 35: 8 36: 1 37: 2 38: 1 39: 128 40: 1 41: 2 42: 1 43: 8 44: 1 45: 2 46: 1 48: 1 49: 2 50: 1 51: 8 52: 1 53: 2 54: 1 55: 128 56: 1 57: 2 58: 1 59: 8 60: 1 61: 2 62: 1 63: 1 95: 32448 *** Line of 0's terminated after 63 lines ============================================= Total Time: 8.96 seconds Graphs of size 16: Number with max value (50): 65520 Percentage with max value: 99.976% FREQUENCY TABLE [value : frequency] 18: 1 19: 1 20: 1 21: 1 22: 1 23: 1 24: 1 25: 1 26: 1 27: 1 28: 1 29: 1 30: 1 31: 1 32: 1 33: 1 50: 65520 *** Line of 0's terminated after 33 lines ============================================= Total Time: 39.69 seconds Graphs of size 17: Number with max value (107): 130456 Percentage with max value: 99.530% FREQUENCY TABLE [value : frequency] 38: 4 39: 2 40: 1 42: 1 43: 2 44: 256 46: 1 47: 32 48: 1 50: 4 51: 2 52: 1 54: 1 55: 2 56: 4 58: 1 59: 32 60: 1 62: 256 63: 2 64: 1 66: 1 67: 2 68: 4 70: 1 71: 1 107: 130456 *** Line of 0's terminated after 71 lines ============================================= Total Time: 44.37 seconds Graphs of size 18: Number with max value (56): 262126 Percentage with max value: 99.993% FREQUENCY TABLE [value : frequency] 20: 1 21: 1 22: 1 23: 1 24: 1 25: 1 26: 1 27: 1 28: 1 29: 1 30: 1 31: 1 32: 1 33: 1 34: 1 35: 1 36: 1 37: 1 56: 262126 *** Line of 0's terminated after 37 lines ============================================= Total Time: 195.44 seconds Graphs of size 19: Number with max value (119): 523136 Percentage with max value: 99.780% FREQUENCY TABLE [value : frequency] 41: 2 42: 1 44: 16 45: 2 46: 1 47: 8 48: 1 49: 512 50: 1 52: 1 53: 2 54: 16 55: 8 56: 1 57: 2 58: 1 60: 1 61: 2 62: 1 63: 8 64: 16 65: 2 66: 1 68: 1 69: 512 70: 1 71: 8 72: 1 73: 2 74: 16 76: 1 77: 2 78: 1 79: 1 119: 523136 *** Line of 0's terminated after 79 lines ============================================= Total Time: 217.02 seconds Graphs of size 20: Number with max value (62): 1048412 Percentage with max value: 99.984% FREQUENCY TABLE [value : frequency] 22: 1 23: 4 24: 1 25: 1 26: 4 27: 64 28: 1 29: 4 30: 1 31: 1 32: 4 33: 1 34: 64 35: 4 36: 1 37: 1 38: 4 39: 1 40: 1 41: 1 62: 1048412 *** Line of 0's terminated after 41 lines ============================================= Total Time: 942.35 seconds Graphs of size 21: Number with max value (131): 2095064 Percentage with max value: 99.900% FREQUENCY TABLE [value : frequency] 46: 1 47: 2 48: 1 50: 1 51: 2 52: 1 54: 1024 55: 2 56: 1 58: 1 59: 2 60: 1 62: 1 63: 2 64: 1 66: 1 67: 2 68: 1 70: 1 71: 2 72: 1 74: 1 75: 2 76: 1024 78: 1 79: 2 80: 1 82: 1 83: 2 84: 1 86: 1 87: 1 131: 2095064 *** Line of 0's terminated after 87 lines ============================================= Total Time: 1038.37 seconds Graphs of size 22: Number with max value (68): 4194282 Percentage with max value: 99.999% FREQUENCY TABLE [value : frequency] 24: 1 25: 1 26: 1 27: 1 28: 1 29: 1 30: 1 31: 1 32: 1 33: 1 34: 1 35: 1 36: 1 37: 1 38: 1 39: 1 40: 1 41: 1 42: 1 43: 1 44: 1 45: 1 68: 4194282 *** Line of 0's terminated after 45 lines =============================================