  We present fixed parameter tractable algorithms for the conflict-free
coloring problem on graphs. Given a graph $G=(V,E)$, \emph{conflict-free
coloring} of $G$ refers to coloring a subset of $V$ such that for every vertex
$v$, there is a color that is assigned to exactly one vertex in the closed
neighborhood of $v$. The \emph{k-Conflict-free Coloring} problem is to decide
whether $G$ can be conflict-free colored using at most $k$ colors. This problem
is NP-hard even for $k=1$ and therefore under standard complexity theoretic
assumptions, FPT algorithms do not exist when parameterised by the solution
size. We consider the \emph{k-Conflict-free Coloring} problem parameterised by
the treewidth of the graph and show that this problem is fixed parameter
tractable. We also initiate the study of \emph{Strong Conflict-free Coloring}
of graphs. Given a graph $G=(V,E)$, \emph{strong conflict-free coloring} of $G$
refers to coloring a subset of $V$ such that every vertex $v$ has at least one
colored vertex in its closed neighborhood and moreover all the colored vertices
in $v$'s neighborhood have distinct colors. We show that this problem is in FPT
when parameterised by both the treewidth and the solution size. We further
apply these algorithms to get efficient algorithms for a geometric problem
namely the Terrain Guarding problem, when parameterised by a structural
parameter.
