On the total domination critical graphs

On the existence problem of the total domination vertex critical graphs

“…Mojdeh and Rad [11] found the following lemma about a total domination vertex critical graph G of order ∆(G) + γ t (G) with δ(G) ≥ 2. Lemma 8 ([11]).…”

“…Hence, for m = 4 or odd m ≥ 3 and for any even ∆ ≥ 2⌊ m−1 2 ⌋, there exists an m-γ t -critical graph of order ∆(G) + m. Now, we want to consider odd ∆. For m = 3, it is known that there is no 3-γ t -critical graph G of order ∆(G) + 3 for ∆(G) = 3, 5, 7 and there is a 3-γ t -critical graph of order ∆(G) + 3 for any odd ∆(G) ≥ 9 [1,11]. For any odd m ≥ 3 and for any odd ∆ ≥ m + 6, let G 2 be a 3-γ t -critical graph of order 12 with ∆(G 2 ) = 9 and δ(G 2 ) ≥ 2 and let G 3 be an (m − 2)-γ t -critical graph of order ∆ + m − 11 with ∆( 3.…”

“…Mojdeh and Rad [11] found 3-γ t -critical graphs of order 3 + ∆(G) for any even ∆(G) and showed that there is no 3-γ t -critical graph G of order 3 + ∆(G) for ∆(G) = 3, 5. Chen and Sohn [1] proved that there is no 3-γ t -critical graph of order ∆(G) + 3 with ∆(G) = 7 and δ(G) ≥ 2.…”

“…A domination and its variations in graph theory have been studied widely and extensively because of its rich applications [2,7,8,6,[9][10][11][12][13][14]. Two books by Haynes, Hedetniemi and Slater [4,5] provide a well written survey on this subject.…”

On the inverse signed total domination number in graphs

“…The k-tuple total domination number (or k-total domination number), denoted γ ×k,t (G), is the smallest number of vertices in a k-tuple total dominating set. (We should note that, k-tuple total domination number is different from total k-distance domination number [6]). Clearly, γ ×1,t (G) = γ t (G).…”