Chimneys in compositions and bargraphs

Motivated by cut points in graph theory, we consider a similar notion in compositions and bargraphs. This is equivalent to counting r-chimneys (a single column extending beyond its immediate neighbours by at least r cells in a bargraph). We establish generating functions for compositions that avoid or count 2-chimneys. Thereafter, in the case of bargraphs we provide two methods for obtaining these generating functions as well as asymptotic estimates for the more general r-chimneys where r >= 1.