Lattices from graph associahedra and subalgebras of the Malvenuto-Reutenauer algebra (Q2217779)

Given a graph \(G\), \textit{M. Carr} and \textit{S. L. Devadoss} [Topology Appl. 153, No. 12, 2155--2168 (2006; Zbl 1099.52001)] defined a polytope \(P_G\) (dubbed the \emph{graph associahedron}) which captures the connectivity properties of \(G\) using the face lattice of \(P_G\). The authors consider two subalgebras of the Malvenuto-Reutenauer Hopf algebra, one constructed by \textit{M. Ronco} [Prog. Math. 299, 339--350 (2012; Zbl 1260.05122)] and another by \textit{N. Reading} [J. Comb. Theory, Ser. A 110, No. 2, 237--273 (2005; Zbl 1133.20027)] (from lattice quotients of weak order) and compare them by considering a poset \(L_G\) arising from the 1-skeleton of \(P_G\).