Graph insertion operads

TL;DR

This paper introduces two new operad structures on multigraphs and pointed oriented multigraphs using combinatorial species, generalizing existing operads and exploring their suboperads and algebraic relationships.

Contribution

It presents novel operad structures on multigraphs and pointed oriented multigraphs, extending known operads and analyzing their suboperads and algebraic connections.

Findings

Defined a canonical operad on multigraphs generalizing Kontsevich-Willwacher operad

Established links between suboperads and classical operads like the commutative operad

Explored the algebraic structure and properties of the new operads

Abstract

Using the combinatorial species setting, we propose two new operad structures on multigraphs and on pointed oriented multigraphs. The former can be considered as a canonical operad on multigraphs, directly generalizing the Kontsevich-Willwacher operad, and has many interesting suboperads. The latter is a natural extension of the pre-Lie operad in a sense developed here and related to the multigraph operad. We study some of the finitely generated suboperads of the multigraph operad and establish links between them and the commutative operad and the commutative magmatic operad.