  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.
