Minimum proper interval graphs

Abstract

A graph G is a proper interval graph if there exists a mapping r from V( G) to the class of closed intervals of the real line with the properties that for distinct vertices v and w we have r(v) ⋔ r(w) ≠ Ø if and only if v and w are adjacent and neither of the intervals r( v), r( w) contain the other. We prove that for every proper interval graph G, | V( G)| ⩾ 2 c( G) - c( K( G)), where c( G) is the number of cliques of G and K( G) is the clique graph of G. If the equality is verified we call G a minimum proper interval graph. The main result is that the restriction to the class of minimum proper interval graphs of clique mapping G → K( G) is a bijection (up to isomorphism) onto the class of proper interval graphs. We find the greatest clique-closed class Σ ( K( Σ) = Σ) contained in the union of the class of connected minimum proper interval graphs and the class of complete graphs. We enumerate the minimum proper interval graphs with n vertices.