COMPOSITIONS OF GRAPHS AND POLYHEDRA .1. BALANCED INDUCED SUBGRAPHS AND ACYCLIC SUBGRAPHS

Let P(G) be the balanced induced subgraph polytope of G. If G has a two-node cutset, then G decomposes into G1 and G2. It is shown that P(G) can be obtained as a projection of a polytope defined by a system of inequalities that decomposes into two pieces associated with G1 and G2. The problem max cx, x is-an-element-of P(G) is decomposed in the same way. This is applied to series-parallel graphs to show that, in this case, P(G) is a projection of a polytope defined by a system with O(n) inequalities and O(n) variables, where n is the number of nodes in G. Also for this class of graphs, an algorithm is given that finds a maximum weighted balanced induced subgraph in O(n log n) time. This approach is also used to obtain composition of facets of P(G). Analogous results are presented for acyclic induced subgraphs.