Article

# The diameter of total domination vertex critical graphs.

• ##### Lucas C. van der Merwe
Discrete Mathematics 01/2004; 286:255-261. DOI: 10.1016/j.disc.2004.05.010
Source: DBLP

ABSTRACT A graph G with no isolated vertex is total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, the total domination number of G v is less than the total domination number of G. These graphs we call t-critical. If such a graph G has total domination number k, we call it k-t-critical. We characterize the connected graphs with minimum degree one that are t-critical and we obtain sharp bounds on their maximum diameter. We calculate the maximum diameter of a k-t-critical graph for k � 8 and provide an example which shows that the maximum diameter is in general at least 5k/3 O(1).

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##### Article: On the existence problem of the total domination vertex critical graphs
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ABSTRACT: The existence problem of the total domination vertex critical graphs has been studied in a series of articles. The aim of the present article is twofold. First, we 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_t-critical graph regardless of the parity of Delta; for m=4 or odd m \ge 3 and for even Delta, an m-gamma_t-critical graph exists if and only if Delta \ge 2 \lfloor \frac{m-1}{2}\rfloor; for m=4 or odd m \ge 3 and for odd Delta, if Delta \ge 2\lfloor \frac{m-1}{2}\rfloor +7, then m-gamma_t-critical graphs exist, if Delta < 2\lfloor \frac{m-1}{2}\rfloor, then m-gamma_t-critical graphs do not exist. The only remaining open cases are Delta = 2\lfloor \frac{m-1}{2}\rfloor +k, k=1, 3, 5. Second, we study these remaining open cases when m=4 or odd m \ge 9. As the previously known result for m = 3, we also show that for Delta(G)= 3, 5, 7, there is no 4-gamma_t-critical graph of order Delta(G)+4. On the contrary, it is shown that for odd m \ge 9 there exists an m-gamma_t-critical graph for all Delta \ge m-1.
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##### Article: On the 3-$\gamma_t$-Critical Graphs of Order $\Delta(G)+3$
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ABSTRACT: Let $\gamma_t(G)$ be the total domination number of graph $G$, a graph $G$ is $k$-total domination vertex critical (or\ just\ $k$-$\gamma_t$-critical) if $\gamma_t(G)=k$, and for any vertex $v$ of $G$ that is not adjacent to a vertex of degree one, $\gamma_t(G-v)=k-1$. Mojdeh and Rad \cite{MR06} proposed an open problem: Does there exist a 3-$\gamma_t$-critical graph $G$ of order $\Delta(G)+3$ with $\Delta(G)$ odd? In this paper, we prove that there exists a 3-$\gamma_t$-critical graph $G$ of order $\Delta(G)+3$ with odd $\Delta(G)\geq 9$.
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##### Article: On the diameter for various types of domination vertex critical graphs
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ABSTRACT: In this paper, we consider various types of domination vertex critical graphs, including total domination vertex critical graphs and independent domination vertex critical graphs and connected domination vertex critical graphs. We provide upper bounds on the diameter of them, two of which are sharp.
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