Composition of Graphs and the Triangle-Free Subgraph Polytope

In this paper, we study a composition (decomposition) technique for the triangle-free subgraph polytope in graphs which are decomposable by means of 3-sums satisfying some property. If a graph G decomposes into two graphs G1 and G2, we show that the triangle-free subgraph polytope of G can be described from two linear systems related to G1 and G2. This gives a way to characterize this polytope on graphs that can be recursively decomposed. This also gives a procedure to derive new facets for this polytope. We also show that, if the systems associated with G1 and G2 are TDI, then the system characterizing the polytope for G is TDI. This generalizes previous results in R. Euler and A.R. Mahjoub (Journal of Comb. Theory series B, vol. 53, no. 2, pp. 235–259, 1991) and A.R. Mahjoub (Discrete Applied Math., vol. 62, pp. 209–219, 1995).