Dotted interval graphs and high throughput genotyping

We introduce a generalization of interval graphs, which we call <i>dotted interval graphs (DIG).</i> A dotted interval graph is an intersection graph of arithmetic progressions (=<i>dotted intervals</i>). Coloring of dotted intervals 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 <i>DIG</i> but that <i>DIG<inf>d</inf></i> graphs, i.e. 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<inf>d</inf></i>, graphs is NP-complete even for <i>d</i> = 2. For any fixed <i>d</i>, we provide a 7/8<i>d</i> approximation for the coloring of <i>DIG<inf>d</inf></i> graphs.

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