Crew, LoganSpirkl, Sophie2022-08-222022-08-222022-04https://doi.org/10.1016/j.jcta.2021.105572http://hdl.handle.net/10012/18592The final publication is available at Elsevier via https://doi.org/10.1016/j.jcta.2021.105572 © 2022. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/For a graph G, its Tutte symmetric function XBG generalizes both the Tutte polynomial TG and the chromatic symmetric function XG. We may also consider XB as a map from the t-extended Hopf algebra G[t] of labelled graphs to symmetric functions. We show that the kernel of XB is generated by vertex-relabellings and a finite set of modular relations, in the same style as a recent analogous result by Penaguiao on the chromatic symmetric function X. In particular, we find one such relation that generalizes the well-known triangular modular relation of Orellana and Scott, and build upon this to give a modular relation of the Tutte symmetric function for any two-edge-connected graph that generalizes the n-cycle relation of Dahlberg and vanWilligenburg. Additionally, we give a structural characterization of all local modular relations of the chromatic and Tutte symmetric functions, and prove that there is no single local modification that preserves either function on simple graphs.enAttribution-NonCommercial-NoDerivatives 4.0 Internationalhttp://creativecommons.org/licenses/by-nc-nd/4.0/Tutte symmetric functionchromatic symmetric functionmodular relationhopf algebravertex-weighted graphsalgebraic combinatoricsArticle