  Let $G = (V,E)$ be a simple, undirected and connected graph. A connected
(total) dominating set $S \subseteq V$ is a secure connected (total) dominating
set of $G$, if for each $ u \in V \setminus S$, there exists $v \in S$ such
that $uv \in E$ and $(S \setminus \lbrace v \rbrace) \cup \lbrace u \rbrace $
is a connected (total) dominating set of $G$. The minimum cardinality of a
secure connected (total) dominating set of $G$ denoted by $ \gamma_{sc} (G)
(\gamma_{st}(G))$, is called the secure connected (total) domination number of
$G$. In this paper, we show that the decision problems corresponding to secure
connected domination number and secure total domination number are NP-complete
even when restricted to split graphs or bipartite graphs. The NP-complete
reductions also show that these problems are w[2]-hard. We also prove that the
secure connected domination problem is linear time solvable in block graphs and
threshold graphs.
