Improper Interval Graphs and the Corresponding Minimal Forbidden Interval Subgraphs

An interval graph is considered improper if and only if it has a representation such that an interval contains another interval. Previously these have been investigated in terms of balance and minimal forbidden interval subgraphs for the class of 1-improper interval graphs. This paper investigates the minimal forbidden interval sub-graphs further, generalizing results to all p-improper interval graphs. It is apparent that there are many different types of possible minimal forbidden subgraphs that fall into four broad categories.