On the existence problem of the total domination vertex critical graphs

Abstract

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 Δ : for even m except m = 4 , there is no m - γ t -critical graph regardless of the parity of Δ ; for m = 4 or odd m ≥ 3 and for even Δ , an m - γ t -critical graph exists if and only if Δ ≥ 2 ⌊ m − 1 2 ⌋ ; for m = 4 or odd m ≥ 3 and for odd Δ , if Δ ≥ 2 ⌊ m − 1 2 ⌋ + 7 , then m - γ t -critical graphs exist, if Δ < 2 ⌊ m − 1 2 ⌋ , then m - γ t -critical graphs do not exist. The only remaining open cases are Δ = 2 ⌊ m − 1 2 ⌋ + 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 Δ ( G ) = 3 , 5 , 7 , there is no 4 - γ t -critical graph of order Δ ( G ) + 4 . On the contrary, it is shown that for odd m ≥ 9 there exists an m - γ t -critical graph for all Δ ≥ m − 1 .