  In this article, we prove a generalization of a theorem of Lisca-Matic to
Stein cobordisms and develop a method for distinguishing certain Stein
cobordisms using rotation numbers. Using these results along with standard
techniques from convex surface theory and classifications of tight contact
structures on certain 3-manifolds due to Honda, we classify the tight contact
structures on a certain class of plumbed 3-manifolds that bound non-simply
connected 4-manifolds. Moreover, we give descriptions of the Stein fillings of
the Stein fillable contact structures.
