MINIMUM PROPER INTERVAL-GRAPHS

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) boolean AND r(w) not equal 0 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)\ greater than or equal to 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 Sigma (K(Sigma) = C) 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.