The Hopf monoid and the basic invariant of directed graphs

Abstract

Aguiar and Ardila defined the Hopf monoid ${\rm GP}$ of generalized permutahedra and showed that it contains many submonoids that correspond to combinatorial objects. They also give a basic polynomial invariant of generalized permutahedra, which then specializes to submonoids. We define the Hopf monoid of directed graphs and show that it also embeds in ${\rm GP}$. The resulting basic invariant coincides with the strict chromatic polynomial of Awan and Bernardi.