Composition of graphs and the triangle free subgraph polytope

Given a graph G which decomposes into two graphs G1 and G2 by means of a 3-sum, we derive a linear system of inequalities that defines the triangle free subgraph polytope from two systems related to G1 and G2. This yields a method to characterize this polytope for graphs that decompose this way. A procedure is given to describe new facets for this polytope. We also show that this system is TDI if the systems for the pieces are also TDI. Some applications of these results will be discussed.