More on discrete convexity

In several recent papers some concepts of convex analysis were extended to discrete sets. This paper is one more step in this direction. It is well known that a local minimum of a convex function is always its global minimum. We study some discrete objects that share this property and provide several examples of convex families related to graphs and to two-person games in normal form

A corrected version of the Duchet Kernel Conjecture

In 1980 Piere Duchet conjectured that odd directed cycles are the only edge minimal kernel-less connected digraphs i.e. in which after the removal of any edge a kernel appears. Although this conjecture was disproved recently by Apartsin, Ferapontova and Gurvich (1996), the following modification of Duchet's conjecture still holds: odd holes (i.e. odd non-directed chordless cycles of length 5 or more) are the only connected graphs which are not kernel-solvable but after the removal of any edge the resulting graph is kernel-solvable. 1 The authors gratefully acknowledge the partial support of DIMACS and ONR (Grants N00014-92-J-1375 and N00014-92-J-4083). Let D = (V; A) be a directed graph (digraph). A subset K ` V of the vertices is called a kernel of D if it is (i) independent (i.e. there are no arcs between its elements), and (ii) dominating (i.e. for every vertex v outside of K there is an arc from a vertex of K to v). A digraph is called kernel-less if it has no kernel. Von Neuman..