On the existence problem of the total domination vertex critical graphs

The existence problem of the total domination vertex critical graphs has been studied in a series of articles we first settle the existence problem with respect to the parities of the total domination number m and the maximum degree Delta for even m except m = 4 there is no m-gamma(iota) critical graph regardless of the parity of Delta for m = 4 or odd m >= 3 and for even Delta an m-gamma(iota)-critical graph exists if and only if Delta >= 2left perpendicular m-1/2 right perpendicular for m = 4 or odd m >= 3 and for odd Delta if Delta >= 2left perpendicular m-1/2 right perpendicular + 7 then m-gamma(iota)-critical graphs exist if Delta < 2left perpendicular m-1/2 right perpendicular then m-gamma(iota)-critical graphs do not exist The only remaining open cases are Delta = 2left perpendicular m-1/2 right perpendicular + k k = 1 3 5 Second we study these remaining open cases when m = 4 or odd m >= 9 As the previously known result for m = 3 we also show that for Delta(G) = 3 5 7 there is no 4-gamma(iota)-critical graph of order Delta (G) + 4 On the contrary it is shown that for odd m >= 9 there exists an m-gamma(iota)-critical graph for all Delta >= m - 1 (C) 2010 Elsevier B V All rights reserved