Classes of interval graphs under expanding length restrictions
P. C. Fishburn, Ronald Graham · Journal of Graph Theory · 1985
Abstract Let C (α) denote the finite interval graphs representable as intersection graphs of closed real intervals with lengths in [1, α]. The points of increase for C are the rational α ≥ 1. The set D (α) = [∩ β>α C (β)]\ C (α) of graphs that appear as soon as we go past α is characterized up to isomorphism on the basis of finite sets E (α) of irreducible graphs for each rational α. With α = p / q and p and q relatively prime, ∣ E (α)∣ is computed for all ( p,q ) with q ⩽ 2 and p = q + 1. When q = 1, E(p ) contains only the bipartite star K 1 , p +2. A lowr bound on ∣ E (α)∣ is given for all rational α.