On Combinatorial Properties of Linear Program Digraphs

An LP-digraph is a directed graph which consists of the vertices and edges of a polytope P directed by a linear function in general position. It is known that an LP-digraph satisfies three necessary conditions: it is acyclic, each face of P has a unique sink (USO), and the Holt-Klee condition holds. These three conditions are not sufficient in general for a directed graph on P to be an LP-digraph. In this paper we survey the relationships between these three condtions and introduce a new necessary condition, the shelling condition. We show that the shelling condition is stronger than a combination of the acyclicity and Holt-Klee conditions for polytopes in dimension at least 4 which are non-simple. In all other cases, we show that it is equivalent to the intersection of these two conditions. Résumé Un LP-digraphe est un graphe orienté comprenant l’ensemble des sommets et des arêtes d’un polytope P , dirigé par une fonction linéaire en position générale. Il est connu qu’un LP-digraphe satisfait à trois conditions nécessaires : il est acyclique, il y a un puit unique dans chaque face de P (USO), et la condition Holt-Klee tient. Ces trois conditions ne sont pas suffisantes en général pour qu’un graphe orienté sur P soit un LP-digraphe. Dans cet article nous faisons une étude des liens entre ces condtions et nous introduisons une nouvelle condition nécessaire, la condition de shelling. Nous montrons que cette condition est plus forte qu’une combinaison des conditions d’acyclicité et d’être un USO pour les polytopes en dimension d’au moins 4 qui ne sont pas simples. En tout autre cas, elle est équivalente à l’intersection de ces deux conditions. Acknowledgments: Research of the first author was supported by NSERC and FQRNT and research of the second author was supported by KAKENHI. The authors would like to thank an anonymous referee for comments on an earlier version of this paper that lead to several important improvements including the much simplified proof of Proposition 6 included here. Les Cahiers du GERAD G–2008–08 1