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The diameter of total domination vertex critical graphs

Discrete Mathematics (Impact Factor: 0.57). 09/2004; 286(3):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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Available from: Michael A. Henning, Jul 09, 2014
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    • "In 2004, Haynes et al [4] introduced the concept of total domination vertex critical. A graph G is total domination vertex critical or just γ t -critical, if for any vertex v of G that is not adjacent to a vertex of degree one, γ t (G − v) < γ t (G). "
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    • "Goddard, Haynes, Henning and van der Merwe [4] introduced the concept of total domination vertex critical. A graph G is k-total domination vertex critical (or just k-γ t -critical) if γ t (G) = k, and for any vertex v of G that is not adjacent to a vertex of degree one, γ t (G− v) = k − 1. "
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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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    • "Lemma 6 ( [2] "
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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.
    Discrete Applied Mathematics 01/2011; 159(1):46-52. DOI:10.1016/j.dam.2010.09.004 · 0.68 Impact Factor
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