Network Realignment Complexes over General Graphs

Network realignment complexes were introduced by Kozlov. We generalise their definition to arbitrary connected base graphs. For a connected graph $G$, we characterise the connected components of the associated network realignment complex $X_G$ and show that $X_G$ admits an $\operatorname{Aut}(G)$-equivariant strong deformation retraction onto the disjoint union of a complete graph and a discrete $\operatorname{Aut}(G)$-space. For the complete base graph $K_n$, we study the metric structure of the network realignment graph $\mathcal{G}_n$ and obtain explicit upper and lower bounds for its diameter. Finally, we prove that $X_n$ is a cubical flag complex and that every automorphism of $X_n$ is induced by a relabelling of the underlying vertex set. In particular, $\operatorname{Aut}(X_n)\cong S_n$ for all $n\geq 5$.