Path-Partitions in Directed Graphs and Posets

Abstract

Some of the results known about chains and antichains in a partially ordered set can be extended to some specific classes of directed graphs. For instance, the “good” antichains, which meet all maximal chains, have two possible extensions: first, the “kernels” of a directed graph appear to be maximal stable sets meeting all maximal paths; second the “stable transversals” are defined to be the maximal stable sets meeting all maximal cliques. Hence some properties of a kernel — resp. a stable transversal —, if there exists one, can generalize theorems in the theory of posets.

Also, in a directed graph, if we consider a partition of the vertex-set into paths μ1, μ2,… which minimizes $\sum\limits_{i} {\min } \{ k,\left| {{{\mu }_{i}}} \right|\} .$ we get interesting properties (exactly as in the theorem of Greene and Kleitman aboutk-saturated chain-partitions of a poset).

In this paper, we survey various results in that direction.