  Directed graphs have long been used to gain understanding of the structure of
semigroups, and recently the structure of directed graph semigroups has been
investigated resulting in a characterization theorem and an analog of Fruct's
Theorem. We investigate four inverse semigroups defined over undirected graphs
constructed from the notions of subgraph, vertex induced subgraph, rooted tree
induced subgraph, and rooted path induced subgraph. We characterize the
structure of the semilattice of idempotents and lattice of ideals of these four
inverse semigroups. Finally, we prove a characterization theorem that states
that every graph has a unique associated inverse semigroup up to isomorphism.
