Dotted interval graphs

We introduce a generalization of interval graphs, which we call Dotted Interval Graphs (DIG). A dotted interval graph is an intersection graph of arithmetic progressions (dotted intervals). Coloring of dotted interval graphs naturally arises in the context of high throughput genotyping. We study the properties of dotted interval graphs, with a focus on coloring. We show that any graph is a DIG, but that <i>DIG<sub>d</sub></i> graphs, that is, DIGs in which the arithmetic progressions have a jump of at most <i>d</i>, form a strict hierarchy. We show that coloring <i>DIG<sub>d</sub></i> graphs is NP-complete even for <i>d</i> = 2. For any fixed <i>d</i>, we provide a 5/6<i>d</i> + <i>o</i>(<i>d</i>) approximation for the coloring of <i>DIG<sub>d</sub></i> graphs. Finally, we show that finding the maximal clique in <i>DIG<sub>d</sub></i> graphs is fixed parameter tractable in <i>d</i>.

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