Fractional Kernels in Digraphs

The in-neighborhood, I(v), of a vertex v in a digraph D=(V, A) is v together with the set of all vertices sending an arc to v, i.e., vertices u such that (u, v) # A. A subset of V is called dominating if it meets I(v) for every v # V. (To avoid confusion, it must be noted that some authors require in the definition meeting every out-neighborhood.) A set of vertices is called independent if no two distinct elements in it are connected by an arc. We shall allow in a digraph pairs of oppositely directed arcs. An arc (u, v) is called irreversible if (v, u) is not an arc of the graph. Like many other combinatorial concepts, these two have fractional counterparts. A non-negative function f on V is called fractionally dominating if u # I(v) f (u) 1 for every vertex v. This requirement can be strengthened, to demand that u # K f (u) 1 for some clique K contained in I(v). (A set K of vertices is a clique if every two vertices in K are connected by at least one arc.) If this holds for every v # V then f is called strongly dominating. A non-negative function f on V is called fractionally independent if u # K f (u) 1 for every clique K. A kernel in D is an independent and dominating set of vertices. A fractional kernel is a function on V which is both fractionally independent and fractionally dominating. In case that it is also strongly dominating, it is Article No. TB971731