  A subset $S\subseteq V$ in a graph $G=(V,E)$ is a total $[1,2]$-set if, for
every vertex $v\in V$, $1\leq |N(v)\cap S|\leq 2$. The minimum cardinality of a
total $[1,2]$-set of $G$ is called the total $[1,2]$-domination number, denoted
by $\gamma_{t[1,2]}(G)$.
  We establish two sharp upper bounds on the total [1,2]-domination number of a
graph $G$ in terms of its order and minimum degree, and characterize the
corresponding extremal graphs achieving these bounds. Moreover, we give some
sufficient conditions for a graph without total $[1,2]$-set and for a graph
with the same total $[1,2]$-domination number, $[1,2]$-domination number and
domination number.
