On shellability for a poset of even subgraphs of a graph

Given a simple graph $G$, a poset of its even subgraphs was firstly considered by S. Choi and H. Park to study the topology of a real toric manifold associated with $G$. S. Choi and the authors extended this to a graph allowing multiple edges, motivated by the work on the pseudograph associahedron of Carr, Devadoss and Forcey. In this paper, we completely characterize the graphs (allowing multiple edges) whose posets of even subgraphs are always shellable. By the result, we also compute the Betti numbers of a real toric manifold corresponding to a path with two multiple edges.