Changes
(1) <math>x(v)+x(N^{out}(v)) \geq 1</math> for every <math>v \in V</math>.
By a result of Rothblum<ref>U. G. Rothblum, ''Characterization of stable matchings as extreme points of a polytope'', Math. Programming Ser. A 54 (1992), 57–67. [http:name="Roth"//dx.doi.org/10.1007/BF01586041 DOI link].</ref> on stable matchings, clique inequalities and inequalities of type (1) describe the convex hull of kernels if the underlying undirected graph of ''D'' is the line graph of a bipartite graph, and every clique is acyclic. The second condition is important: the figure on the left below represents an orientation of the line graph of <math>K_{3,3}</math> (each 3-clique is a cycle) where there is no kernel, but the all-1/3 vector satisfies the inequalities of type (1).
The figure on the right is an example by Frederic Meunier showing an oriented simple bipartite graph with a vector on the nodes that satisfies the inequalities of type (1) but is not a convex combination of kernels, since each kernel has size 3.
One trivial case is when ''D'' is an acyclic digraph: in this case there is one kernel, and its characteristic vector is the only vector satisfying (1) and <math>x(u)+x(v) \leq 1</math> for every <math>uv \in A</math>.
If ''D'' is the union of two transitive acyclic subdigraphs, then a kernel always exists (in other words, any two partial orders have a stable common antichain, see Fleiner <ref name="Fl"/>). However, Fleiner <ref name="Fl"/> also showed that deciding whether there is a stable common antichain containing a given element is NP-complete, so a compact polyhedral description of kernels is unlikely in this case.
==References==
<references>
<ref name="Roth">U. G. Rothblum, ''Characterization of stable matchings as extreme points of a polytope'', Math. Programming Ser. A 54 (1992), 57–67. [http://dx.doi.org/10.1007/BF01586041 DOI link].</ref>
<referencesref name="Fl">T. Fleiner, ''Stable and crossing structures'', Ph.D. dissertation, [http://www.renyi.hu/~fleiner/dissertation.pdf author link].</ref> </references>
[[Category:Stable matchings and kernels]]
