  It is commonly believed that models defined on a closed one-dimensional
manifold cannot give rise to topological order. Here we construct
frustration-free Hamiltonians which possess both symmetry protected topological
order (SPT) on the open chain {\it and} multiple ground state degeneracy (GSD)
that is unrelated to global symmetry breaking on the closed chain. Instead of
global symmetry breaking, there exists a {\it local} symmetry operator that
commutes with the Hamiltonian and connects the multiple ground states,
reminiscent of how the topologically distinct ground states of the toric code
are connected by various winding operators. Our model solved on an open chain
demonstrates symmetry fractionalization as an indication of SPT order and on a
general graph the GSD can be shown to scale with the first Betti number - a
topological invariant that counts the number of independent cycles or one
dimensional holes of the graph.
