Fixed-Parameter Complexity of Minimum Profile Problems.
G. Gutin, S. Szeider and A. Yeo
In: 2nd International Workshop on Parameterized and Exact Computation, 13 Sept - 15 Sept, 2006, Zurich, Switzerland.

## Abstract

An ordering of a graph $G=(V,E)$ is a one-to-one mapping $\alpha:\ V\dom \{1,2,\ldots, |V|\}.$ The profile of an ordering $\alpha$ of $G$ is $\prf_{\alpha}(G)=\sum_{v\in V}(\alpha(v)-\min\{\alpha(u):\ u\in N[v]\})$; here $N[v]$ denotes the closed neighborhood of~$v$. The profile $\prf(G)$ of $G$ is the minimum of $\prf_{\alpha}(G)$ over all orderings $\alpha$ of $G$. It is well-known that $\prf(G)$ equals the minimum number of edges in an interval graph $H$ that contains $G$ as a subgraph. We show by reduction to a problem kernel of linear size 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 $k$. Since $|V|-1$ is a tight lower bound for the profile of a connected graph $G=(V,E)$, the parameterization above the guaranteed value $|V|-1$ is of particular interest.