Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the 3-$\gamma_t$-Critical Graphs of Order $\Delta(G)+3$   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 $\Delta(G)\geq
9$.

8
The categories of this paper are: hep-ex
35
The categories of this paper are: mtrl-th
14
The categories of this paper are: math-ph
9
The categories of this paper are: nlin
29
The categories of this paper are: funct-an
17
The categories of this paper are: q-fin
30
The categories of this paper are: q-alg
28
The categories of this paper are: alg-geom
33
The categories of this paper are: plasm-ph
31
The categories of this paper are: ao-sci
5
The categories of this paper are: gr-qc
36
The categories of this paper are: cmp-lg
15
The categories of this paper are: nucl-ex
1
The categories of this paper are: cs
0
The categories of this paper are: hep-ph
37
The categories of this paper are: supr-con