  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.
