Fixed-Parameter Complexity of Minimum Profile Problems

Abstract

An ordering of a graph G=(V,E) is a one-to-one mapping α: V →{1,2,..., |V|}. The profile of an ordering α of G is prf _α (G)=∑ _v ∈ V (α(v)– min {α(u): u ∈N[v]}); here N[v] denotes the closed neighborhood of v. The profile prf(G) of G is the minimum of prf _α (G) over all orderings α 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.