The Interval Count of a Graph
Rochelle Leibowitz, Susan F. Assmann, G. W. Peck · SIAM Journal on Algebraic and Discrete Methods · 1982
The interval count of an interval graph G is the minimum number of different interval sizes needed to represent the vertices of G, where two vertices are adjacent if and only if their intervals intersect. We show that if G is an interval graph and for some vertex x, $G - \{ x \}$ has interval count one, then G has interval count two or less. We also show how to construct examples of interval graphs where the interval count of G exceeds that of $G - \{ x \}$ by at least two when the latter number is two or more.