  Let $G=(V,E)$ be a graph. An ordering of $G$ is a bijection $\alpha: V\dom
\{1,2,..., |V|\}.$ For a vertex $v$ in $G$, its closed neighborhood is
$N[v]=\{u\in V: uv\in E\}\cup \{v\}.$ The profile of an ordering $\alpha$ of
$G$ is $\prf_{\alpha}(G)=\sum_{v\in V}(\alpha(v)-\min\{\alpha(u): u\in
N[v]\}).$ The profile $\prf(G)$ of $G$ is the minimum of $\prf_{\alpha}(G)$
over all orderings $\alpha$ of $G$. It is well-known that $\prf(G)$ is the
minimum number of edges in an interval graph $H$ that contains $G$ is a
subgraph. Since $|V|-1$ is a tight lower bound for the profile of connected
graphs $G=(V,E)$, the parametrization above the guaranteed value $|V|-1$ is of
particular interest. We show that deciding whether the profile of a connected
graph $G=(V,E)$ is at most $|V|-1+k$ is fixed-parameter tractable with respect
to the parameter $k$. We achieve this result by reduction to a problem kernel
of linear size.

Data Structures and Algorithms;Discrete Mathematics