On the existence of kernels and h-kernels in directed graphs

On the existence of kernels and h-kernels in directed graphs

Galeana-Sanchez, H., On the existence of kernels and h-kernels in directed graphs, Discrete Mathematics 110 (1992) 2.51-255. A directed graph D with vertex set V is called cyclically h-partite (h > 2) provided one can partition V = V, + V, +. . + V,_, so that if (u, u) is an arc of D then u E V,, and u E v+, (notation mod h). In this communication we obtain a characterization of cyclically h-pa...

Computing Kernels in Directed Bichromatic Graphs

A note on kernels in h-perfect graphs

Boros and Gurvich [3] showed that every clique-acyclic superorientation of a perfect graph has a kernel. We prove the following extension of their result: if G is an h-perfect graph, then every clique-acyclic and odd-hole-acyclic superorientation of G has a kernel. We propose a conjecture related to Scarf’s Lemma that would imply the reverse direction of the Boros-Gurvich theorem without relyin...

Kernels and Regularization on Graphs

We introduce a family of kernels on graphs based on the notion of regularization operators. This generalizes in a natural way the notion of regularization and Greens functions, as commonly used for real valued functions, to graphs. It turns out that diffusion kernels can be found as a special case of our reasoning. We show that the class of positive, monotonically decreasing functions on the un...

Recent problems and results about kernels in directed graphs

Let G be a directed graph. Its vertex-set will be denoted by X, and its arcs (or ‘directed edges’) are a subset of the Cartesian product X x X. A kernel of G is a subset S of X which is ‘stable’ (independent, i.e.: a vertex in S has no successor in S) and ‘absorbant’ (dominating, i.e. a vertex not in S has a successor in S). This concept has found many applications, for instance in cooperative ...