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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
$\gamma$$t$-critical. If such a graph $G$ has total domination
number $k$, we call it $k-\gamma$$t$-critical. We verify an open
problem of $k-\gamma t$-critical graphs and obtain some results on
the characterization of total domination critical graphs of order
$n = \Delta(G)(\gamma t(G) - 1) + 1$.
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