Interval Graphs with Containment Restrictions

An interval graph is proper iff it has a representation in which no interval contains another. Fred Roberts characterized the proper interval graphs as those containing no induced star $K_{1,3}$. Proskurowski and Telle have studied $q$-proper graphs, which are interval graphs having a representation in which no interval is properly contained in more than $q$ other intervals. Like Roberts they found that their classes of graphs where characterized, each by a single minimal forbidden subgraph. This paper initiates the study of $p$-improper interval graphs where no interval contains more than $p$ other intervals. This paper will focus on a special case of $p$-improper interval graphs for which the minimal forbidden subgraphs are readily described. Even in this case, it is apparent that a very wide variety of minimal forbidden subgraphs are possible.