The projected faces property and polyhedral relations

Margot (1994) in his doctoral dissertation studied extended formulations of combinatorial polytopes that arise from “smaller” polytopes via some composition rule. He introduced the “projected faces property” of a polytope and showed that this property suffices to iteratively build extended formulations of composed polytopes. For the composed polytopes, we show that an extended formulation of the type defined by Margot is always possible only if the smaller polytopes have the projected faces property. Therefore, this produces a characterization of the projected faces property. Affinely generated polyhedral relations were introduced by Kaibel and Pashkovich (Optima 85:2–7, 2011) to construct extended formulations for the convex hull of the images of a point under the action of some finite group of reflections. In this paper we prove that the projected faces property and affinely generated polyhedral relation are equivalent conditions.SCOPUS: ar.jinfo:eu-repo/semantics/publishe

Articulation sets in linear perfect matrices I: forbidden configurations and star cutsets

AbstractA (0, 1) matrix is linear if it does not contain a 2 × 2 submatrix of all ones. In these two papers we deal with perfect graphs whose clique-node incidence matrix is linear. We first study properties of some subgraphs that contain odd holes. We then prove that a graph whose clique-node incidence matrix is linear but not totally unimodular contains a node v such that the removal of v and all its neighbors disconnects the graph. These results yield a proof of the strong perfect graph conjecture for this class of graphs