Valid inequalities for a single constrained 0-1 MIP set intersected with a conflict graph

We consider a mixed integer set which results from the intersection of a single constrained mixed 0-1 set with the vertex packing set. This set arises as a subproblem of more general mixed integer problems. We describe families of strong valid inequalities that consider the structure of the simple mixed integer set and the vertex packing set simultaneously. In particular, a class of valid inequalities that generalizes the well-known mixed integer rounding inequalities to the case where incompatibility between binary variables is considered, is derived. We discuss the separation problems associated to those valid inequalities and present a preliminary computational experiment.