Optimization Problems in Dotted Interval Graphs

The class of D-dotted interval (D-DI) graphs is the class of intersection graphs of arithmetic progressions with jump (common difference) at most D. We consider various classical graphtheoretic optimization problems in D-DI graphs of arbitrarily, but fixed, D. We show that Maximum Independent Set, Minimum Vertex Cover, and Minimum Dominating Set can be solved in polynomial time in this graph class, answering an open question posed by Jiang (Information Processing Letters 98(1):29–33, 2006). We also show that Minimum Vertex Cover can be approximated within a factor of (1 + ε) for any ε > 0 in linear time. This algorithm generalizes to a wide class of deletion problems including the classical Minimum Feedback Vertex Set and Minimum Planar Deletion problems. Our algorithms are based on classical results in algorithmic graph theory and new structural properties of D-DI graphs that may be of independent interest.