  The geometric vertex decomposability property for polynomial ideals is an
ideal-theoretic generalization of the vertex decomposability property for
simplicial complexes. Indeed, a homogeneous geometrically vertex decomposable
ideal is radical and Cohen-Macaulay, and is in the Gorenstein liaison class of
a complete intersection (glicci). In this paper, we initiate an investigation
into when the toric ideal $I_G$ of a finite simple graph $G$ is geometrically
vertex decomposable. We first show how geometric vertex decomposability behaves
under tensor products, which allows us to restrict to connected graphs. We then
describe a graph operation that preserves geometric vertex decomposability,
thus allowing us to build many graphs whose corresponding toric ideals are
geometrically vertex decomposable. Using work of Constantinescu and Gorla, we
prove that toric ideals of bipartite graphs are geometrically vertex
decomposable. We also propose a conjecture that all toric ideals of graphs with
a square-free degeneration with respect to a lexicographic order are
geometrically vertex decomposable. As evidence, we prove the conjecture in the
case that the universal Gr\"obner basis of $I_G$ is a set of quadratic
binomials. We also prove that some other families of graphs have the property
that $I_G$ is glicci.
