One-node cutsets and the dominating set polytope

In this paper we study a composition (decomposition) technique for the dominating set polytope in graphs which are decomposable by one-node cutsets. If G decomposes into G1 and G2, we show that the dominating set polytope of G can be described from two linear systems related to G1 and G2. This gives a way to characterize this polytope for classes of graphs that can be recursively decomposed. This also gives a procedure to describe facets for this polytope. Application of these techniques is discussed for the class of the cactus.ou