Composition of graphs and polyhedra I : Balanced induced subgraphs and acyclic subgraphs
Barahona, Francisco; Mahjoub, Ali Ridha (1994), Composition of graphs and polyhedra I : Balanced induced subgraphs and acyclic subgraphs, SIAM Journal on Discrete Mathematics, 7, 3, p. 344-358. http://dx.doi.org/10.1137/S0895480190182666
TypeArticle accepté pour publication ou publié
Journal nameSIAM Journal on Discrete Mathematics
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Abstract (EN)Let $P( G )$ be the balanced induced subgraph polytope of $G$. If $G$ has a two-node cutset, then $G$ 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 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.
Subjects / Keywordsacyclic subgraphs; compact systems; balanced subgraphs; composition of polyhedra; Polyhedral combinatorics
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