Directeds-t numberings, Rubber bands, and testing digraphk-vertex connectivity

Abstract

LetG=(V, E) be a directed graph andn denote |V|. We show thatG isk-vertex connected iff for every subsetX ofV with |X| =k, there is an embedding ofG in the (k−1)-dimensional spaceRk−1,f∶V→Rk−1, such that no hyperplane containsk points of {f(v)|v∈V}, and for eachv∈V−X, f(v) is in the convex hull of {f(w)| (v, w)∈E}. This result generalizes to directed graphs the notion of convex embeddings of undirected graphs introduced by Linial, Lovasz and Wigderson in “Rubber bands, convex embeddings and graph connectivity”,Combinatorica8 (1988), 91–102.