Classes of interval graphs under expanding length restrictions

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>αC(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 K¹, p_+2. A lowr bound on ∣E(α)∣ is given for all rational α.