Open-interval graphs versus closed-interval graphs

Abstract

A graph G = ( V, E) is said to be represented by a family F of nonempty sets if there is a bijection f: V → F such that uv ϵ E if and only if f( u)∩ f( v) ≠ Ø. It is proved that if G is a Countable graph then G can be represented by open intervals on the real line if and only if G can be represented by closed intervals on the real line, however, this is no longer true when G is an uncountable graph. Similar results are also proved when intervals are required to have unit length.