Minimal proper interval completions

Given an arbitrary graph G=(V,E) and a proper interval graph H=(V,F) with E⊆F we say that H is a proper interval completion of G. The graph H is called a minimal proper interval completion of G if, for any sandwich graph H′=(V,F′) with E⊆F′⊂F, H′ is not a proper interval graph. In this paper we give a O(n+m) time algorithm computing a minimal proper interval completion of an arbitrary graph. The output is a proper interval model of the completion.