Given a nontrivial homogeneous ideal I⊆k[x1,x2,…,xd], a problem of great recent interest has been the comparison of the rth ordinary power of I and the mth symbolic power I(m). This comparison has been undertaken directly via an exploration of which exponents m and r guarantee the subset containment I(m)⊆Ir and asymptotically via a computation of the resurgence ρ(I), a number for which any m/r>ρ(I) guarantees I(m)⊆Ir. Recently, a third quantity, the symbolic defect, was introduced; as It⊆I(t), the symbolic defect is the minimal number of generators required to add to It in order to get I(t). We consider these various means of comparison when I is the edge ideal of certain graphs by describing an ideal J for which I(t)=It+J. When I is the edge ideal of an odd cycle, our description of the structure of I(t) yields solutions to both the direct and asymptotic containment questions, as well as a partial computation of the sequence of symbolic defects.