  A subset $D$ of vertices of a graph $G$ is a total dominating set if every
vertex of $G$ is adjacent to at least one vertex of $D$. The total dominating
set $D$ is called a total co-independent dominating set if the subgraph induced
by $V-D$ is edgeless and has at least one vertex. The minimum cardinality of
any total co-independent dominating set is the total co-independent domination
number of $G$ and is denoted by $\gamma_{t,coi}(G)$. In this work we study some
complexity and combinatorial properties of $\gamma_{t,coi}(G)$. Specifically,
we prove that deciding whether $\gamma_{t,coi}(G)\le k$ for a given integer $k$
is an NP-complete problem and give several bounds on $\gamma_{t,coi}(G)$. Also,
since any total co-independent dominating set is also a total dominating set,
we characterize all the trees having equal total co-independent domination
number and total domination number.
