 
Summary: Dotted Interval Graphs and High Throughput Genotyping
Yonatan Aumann \Lambda Moshe Lewenstein \Lambda Oren Melamud \Lambda Ron Pinter y
Zohar Yakhini z
July 4, 2004
Abstract
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 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 DIG but that DIG d graphs, i.e. DIGs in which the arithmetic progressions have a
jump of at most d, form a strict hierarchy. We show that coloring DIG d graphs is NPcomplete
even for d = 2. For any fixed d, we provide a 7
8 d approximation for the coloring of DIG d graphs.
1 Introduction
Overview. Interval graphs have been extensively studied and have many applications [5]. A
graph is an interval graph if the nodes correspond to intervals on the real axis, and there is a
edge between two nodes iff their corresponding intervals overlap. We introduce a generalization of
interval graphs, which we call Dotted Interval Graphs (DIG), in which instead of solid intervals we
consider ``Dotted Intervals'', i.e. segments of a ``dotted line''. Formally, a Dotted Interval (DI) is
an arithmetic progressions of integer values. Thus, the nodes of a dotted interval graph correspond
