We initiate the study of a new parameterization of graph problems. In a multiple interval representation of a graph, each vertex is associated to at least one interval of the real line, with an edge between two vertices if and only if an interval associated to one vertex has a nonempty intersection with an interval associated to the other vertex. A graph on n vertices is a k-gap interval graph if it has a multiple interval representation with at most n+k intervals in total. In order to scale up the nice algorithmic properties of interval graphs (where k=0), we parameterize graph problems by k, and find FPT algorithms for several problems, including Feedback Vertex Set, Dominating Set, Independent Set, Clique, Clique Cover, and Multiple Interval Transversal. The Coloring problem turns out to be W[1]-hard and we design an XP algorithm for the recognition problem.
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