Unit Interval Graphs of Open and Closed Intervals 1 Dieter Rautenbach and Jayme L. Szwarcfiter 2 1 Institut f¨ ur Optimierung und Operations Research, Universit¨at Ulm, 3 Ulm, Germany, dieter.rautenbach@uni-ulm.de 4 2 Instituto de Matem´ atica, NCE, and COPPE, Universidade Federal do Rio de Janeiro, 5 Rio de Janeiro, RJ, Brazil, jayme@nce.ufrj.br 6 Abstract 7 We give two structural characterizations of the class of finite intersection graphs of the 8 open and closed real intervals of unit length. This class is a proper superclass of the well 9 known unit interval graphs. 10 Keywords: intersection graph; interval graph; proper interval graph; unit interval graph 11 MSC 2010 classification: 05C62, 05C75 12 1 Introduction 13 A graph is an interval graph if one can assign an interval of the real line to each of its vertices 14 in such a way that two vertices are adjacent exactly if the corresponding intervals intersect, i.e. 15 interval graphs are the intersection graphs of real intervals. Interval graphs and some subclasses 16 like proper/unit interval graphs have well studied structural [2, 5, 8, 15] as well as algorithmic 17 [3, 4, 9, 10] properties and occur in many applications such as archaeology [12], scheduling [13], 18 and physical mapping of DNA [7, 11]. 19 Many references do actually not specify whether open or closed intervals are used for the 20 intersection representation of interval graphs. The reason for this might be that using either 21 only open intervals or only closed intervals leads to the same classes of finite graphs [6]. As we 22 will observe in Proposition 1 below, even allowing the use of open as well as closed intervals 23 within one representation does not lead to a larger class of graphs. This changes for the well 24 known subclass of unit interval graphs where the intervals are required to be of the same length. 25 Tucker [16] made an analogous observation for unit circular arc graphs. As shown in Proposition 26 4 below, the claw K 1,3 , which is the only minimal interval graph that is not also a unit interval 27 graph, has an intersection representation using open as well as closed intervals of unit length. 28 The purpose of the present paper is to give structural characterizations of the class of inter- 29 section graphs of the open and closed real intervals of unit length. In Section 2 we collect basic 30 terminology, definitions, and results. In Section 3 we provide the structural characterizations. 31 1