  A path in an(a) edge(vertex)-colored graph is called \emph{a conflict-free
path} if there exists a color used on only one of its edges(vertices). An(A)
edge(vertex)-colored graph is called \emph{conflict-free (vertex-)connected} if
there is a conflict-free path between each pair of distinct vertices. We call
the graph $G$ \emph{strongly conflict-free connected }if there exists a
conflict-free path of length $d_G(u,v)$ for every two vertices $u,v\in V(G)$.
And the \emph{strong conflict-free connection number} of a connected graph $G$,
denoted by $scfc(G)$, is defined as the smallest number of colors that are
required to make $G$ strongly conflict-free connected. In this paper, we first
investigate the question: Given a connected graph $G$ and a coloring $c: E(or\
V)\rightarrow \{1,2,\cdots,k\} \ (k\geq 1)$ of the graph, determine whether or
not $G$ is, respectively, conflict-free connected, vertex-conflict-free
connected, strongly conflict-free connected under coloring $c$. We solve this
question by providing polynomial-time algorithms. We then show that it is
NP-complete to decide whether there is a k-edge-coloring $(k\geq 2)$ of $G$
such that all pairs $(u,v)\in P \ (P\subset V\times V)$ are strongly
conflict-free connected. Finally, we prove that the problem of deciding whether
$scfc(G)\leq k$ $(k\geq 2)$ for a given graph $G$ is NP-complete.
