A short proof that 'proper = unit'

A short proof is given that the graphs with proper interval representations are the same as the graphs with unit interval representations. An graph is an interval graph if its vertices can be assigned intervals on the real line so that vertices are adjacent if and only if the corresponding intervals intersect; such an assignment is an interval representation. When the intervals have the same length, we have a unit interval representation. When no interval properly contains another, we have a proper interval representation. The unit interval graphs and proper interval graphs are the interval graphs having unit interval or proper interval representations, respectively. Since no interval contains another of the same length, every unit interval graph is a proper interval graph. Roberts [1] proved that also every proper interval graph is a unit interval graph; the two classes are the same. He proved this as part of a characterization of unit interval graphs as the interval graphs with no induced subgraph isomorphic to the “claw” K1,3 (it is immediate that the condition is necessary). Roberts used a version of the Scott-Suppes [2] characterization of semiorders to prove that claw-free interval graphs are unit interval graphs. By eschewing the trivial implication that unit interval graphs are proper interval graphs and instead going from “claw-free” to “proper” to “unit” among the interval graphs, we obtain a short self-contained proof. In the language of partial orders, our proof also characterizes the semiorders among the interval orders. A partial order is an interval order if its elements can be assigned intervals on the real line so that x < y if and only if the interval assigned to x is completely to the left of the interval assigned to y. A partial order is a semiorder if its elements can be assigned numbers so that x < y if and only if the number assigned to y exceeds the number assigned to x by more than 1. The poset 1+3 is the poset consisting of two disjoint chains of sizes 3 and 1. The semiorders are precisely the interval orders that do not contain 1 + 3. †Research supported in part by NSA/MSP Grant MDA904-93-H-3040. Running head: Proper = Unit AMS codes: 05C75, 06A07