Compositions of Graphs and Polyhedra I: Balanced Induced Subgraphs and Acyclic Subgraphs

Let $P(G)$ be the balanced subgraph polytope of $G$. If $G$ has a two-node cutset, the $G$ decomposes into $G_1$ and $G_2$. 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 $G_1$ and $G_2$. The problem max $cx, x \in P(G)$ 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 case of graphs, an algorithm is given that finds a maximum weighted balanced 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.